SFUSD Mathematics Core Curriculum Development Project€¦ · Students will perform dilations on...

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SFUSD Mathematics Core Curriculum, Grade 10, Unit G.4: Similarity, 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.4 Similarity

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Enlarging Triangles 1 per student Straightedges or rulers

3 Lesson Series 1 Similarity Outside (2 pages) Similarity Outside Reflection Blowing up Shapes! Stick Figure Dilations (2 pages)

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

Chalk String (optional) Measuring tape or meter sticks (optional)

1 Apprentice Task Walk it Out Walk it Out Some More What’s Our Distance After We Walk

1 per pair 1 per pair 1 per pair

4 Lesson Series 2 Let’s All Walk it Out – Parts 1 and 2 (2 pages) Was That Even Possible? Similarity Transformation Angle-Angle Similarity (2 pages)

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

1 Expert Task Proving Proportionality 1 per pair Poster paper and markers

3 Lesson Series 3 Floodlights Sample Responses to Discuss (3 pages) How Did You Work?

1 per student 1 per pair 1 per student

Rulers and protractors (optional) Calculators (optional)

1 Milestone Task Lessons from Abroad (2 pages) 1 per student

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Unit Overview

Big Idea

Similarity can be thought of as a relationship between shapes formed through the use of dilations.

Unit Objectives

● Students will perform dilations on points to construct shapes. ● Students will make statements about the relationship of the original points/shapes to the image created. ● Students will perform the same dilations on a coordinate plane and make statements about the relationship of the resulting coordinate points. ● Students will prove theorems involving similarity. ● Students will solve area problems using what they know about similar shapes.

Unit Description

Students will start with the basic idea of what a dilation is (in respect to the other transformations that they previously studied) and how dilations can create similar shapes. The unit continues with students being more precise (via construction and coordinate geometry) about what specific properties are similar between the original shapes and created images in terms of sides and angles. Lastly, the unit finishes off with the practice of extending the idea of similarity to help prove other theorems involving the Pythagorean Theorem.

CCSS-M Content Standards

G-SRT Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor: G.SRT.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. G.SRT.1b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity

transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

G.SRT.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

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Prove theorems involving similarity G.SRT.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the

Pythagorean Theorem proved using triangle similarity. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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Progression of Mathematical Ideas Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

Students work with transformations informally in Unit 8.3. In Unit G.1 students are introduced to constructions and in Unit G.2, students work more formally with rigid motions (reflections, rotations, and translations) and develop an understanding of congruence as a series of rigid motions that transform one shape into another.

Similarity can be thought of as a relationship between shapes formed through the use of a dilation in addition to rigid motions. The ratios of the lengths of parts of similar figures are equal to the scale factor of the dilation, and the angles remain congruent. Similarity can also be useful in solving problems that involve finding missing lengths of geometric figures, including finding the area of a geometric shapes.

Similarity is the basis for the development of right triangle trigonometry in Unit G.7.

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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 3 Days 1 Day 4 Days 1 Day 3 Days 1 Day

Total: 14 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

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Entry Task Enlarging Triangles

Apprentice Task Walk it Out

Expert Task Proving Proportionality

Milestone Task Lessons from Abroad

CCSS-M Standards

G.SRT.1a, G.SRT.1b G.SRT.2, G.SRT.3 G.SRT.4 G.SRT.2, G.SRT.5

Brief Description of Task

In this task, students will create a dilated shape from a given shape and then make comparisons between the corresponding side lengths. The purpose of the task is to introduce students to the idea of creating similar shapes by dilating side lengths.

In this task, students will start at a point P and move along different paths to points A and B, where the distance in which they move can be doubled, tripled, etc to find A’ and B’. They will then compare the distance between where they end up from case to case (AB compared to A’B’), seeing that the distances will be proportional and parallel (i.e., the slope will be equivalent).

In this task, students are given two triangles and use transformations (including dilations) to show that the triangles can be lined up to show the third side is parallel which shows a dilation exists which shows they are similar.

In this task, students will be able to make a model of a real-world situation, draw diagrams to help with solving a problem, and identify similar triangles and use their properties to solve problems.

Source SFUSD Teacher Created SFUSD Teacher Created SFUSD Teacher Created CPM CCG Lesson 3.2.6

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards

G.SRT.1a, G.SRT.1b, G.SRT.2 G.SRT.3, G.SRT.4 G.SRT.5

Brief Description of Lessons

Introduction to Dilations as a Fourth Transformation Similarity Outside Blowing Up Shapes with Dilations Stick Figure Dilations

Introduction to Similarity and the Properties of Figures that are Similar Let’s ALL Walk it Out Was that Even Possible? Similarity Transformation Angle-Angle Similarity Shortcut

Floodlights Formative Assessment Lesson Evaluating sample responses about solving problems using similarity (aka proportional reasoning)

Sources SFUSD Teacher Created SFUSD Teacher Created FAL “Floodlights”

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Entry Task

Enlarging Triangles

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● This task will help students to think about what they already know

about making a triangle bigger. ● Through this task, students should be able to think about the

relationship between extending lines of an original shape to create a larger shape, which is the transformation known as dilation.

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

● Students may not know that to “extend” a line on a coordinate plane is to maintain the same slope but to grow its length.

● Students may have a difficult time seeing the nested triangles as two separate triangles. They may see a triangle and a trapezoid instead.

● Students may extend each leg of the triangle and not complete a new triangle. For example:

VS.

Launch: Students may start with a Do Now about how to make something twice as big. For example, “draw a shape that is twice as big as the triangle below” or maybe also “draw a shape that is half as big as the triangle below.”

Students should recognize that the resulting shape would also be a triangle and one feature of the triangle (i.e., side lengths, area, perimeter, etc.) would be twice/half as big. During: As students complete the task, be sure to ask questions to elicit their prior knowledge about how they know that AB was extended to be twice as long. This is not supposed to be a rigorous task but rather help to uncover some possible misconceptions. Closure/Extension: A good closure problem after students have debriefed what they noticed in this task could be to ask them to describe “Shrinking Triangles” and replace the phrase “extend to be twice as long” with the phrase “reduce to be twice as short.” For other extensions, students could try other fractions to make it more computationally challenging.

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Enlarging Triangles

How will students do this?

Focus Standards for Mathematical Practice: 6. Attend to precision. 7. Look for and make use of structure.

Structures for Student Learning: Academic Language Support:

Vocabulary: extend, triangle, twice as big, half as big

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

Students should work in groups so that they can compare their results of their sketches.

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Lesson Series #1 Lesson Series Overview: The purpose of Lesson Series #1 is to help students connect how the transformation of dilation leads to a relationship between the original and created shapes, which is known as similarity. Students are given tasks that practice transforming lines and shapes by performing dilations and then asked to compare the features of the two resulting shapes. CCSS-M Standards Addressed: G.SRT.1a, G.SRT.1b, G.SRT.2 Time: 3 days

Lesson Overview - Day 1 Resources

Description of Lesson: Day 1 consists of an activity called “Similarity Outside” where students explore how the extended side lengths correspond to the angles within the triangles. In step 7, make sure that the Team Captain starts from point A and continues along ray VA, and that the Facilitator starts from point B and continues along ray VB. After doing the task outside, have students answer the reflection questions. In debriefing the activity, here are some questions to ask:

• Does it matter what the angles in your triangle are? • Which triangles were easiest to draw on your paper? Which were most difficult? • Are the triangles still proportional if the angles are different? • How close were your predictions? • Why does setting up proportions work?

Notes: There are some instructions that students need to pay attention to while in the classroom as well as some while they are outside. The blacktop is a good area to complete this activity because students will need to use chalk to write on the ground. The instructions ask students to use a “step” as the unit of measurement. As long as the same people walk along the same sides using fairly consistent steps, this is adequate. If this proves to be problematic, groups could use measuring tape or meter sticks to measure the sides using a standard measure such as feet or meters.

Similarity Outside Task Similarity Outside Roles Similarity Outside Reflection

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Description of Lesson: Day 2 consists of a task called “Blowing Up Shapes” where students explore how a point of dilation affects the resulting image even if the dilation factor stays constant. Have different students in each group fill in the two blanks on the handout with either twice, three times, or four times. These will all fit on the given grid. Notes: Students may want to use graph paper to complete this task if the given graph on the handout is not big enough for them.

Blowing Up Shapes


Description of Lesson: Day 3 consists of a task called “Stick Figure Dilation” where students use what they’ve been learning about dilations to try to create specific images. The task assumes that the dilation factor can be anything, but the point of dilation matters in how to correctly position the dilated image. Notes: The last question is open-ended enough so that students can explain as much as they have learned about dilations.

Stick Figure Dilation

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Apprentice Task Walk it Out



Math Objectives: ● Students will use different strategies to find the distance of diagonal

line segments on a graph. ● Students will notice if two line segments, AP and BP, are connected to

a fixed point P, then the distance of AP, BP, and AB will always be proportional.

CCSS-M Standards Addressed: G.SRT.2, G.SRT.3 Potential Misconceptions:

● The big misconception to look out for here is the idea of distance along a diagonal line versus distance along the grid. Some kids will know that they can use the Pythagorean Theorem (or distance formula), but some will think they can count diagonal distance the same way they do distance along the grid-lines (+1 for every time you hit a grid line). It is important that they see the difference and keep it in mind when looking for points that double or triple a given distance.

Launch: Students may start with the following Do Now. This will help students review their methods of finding the distance of a diagonal line on a grid and help them think about what it means to create a line with a given distance.

Note: Students may come up with different strategies. If they are familiar with the Pythagorean Theorem, there strategies for #1 can be straightforward (10 units). However, their attempts at making diagonal line half as long would require more thinking. Push them to think about the distance traveled if they were only to move horizontally and vertically. This may help them with #2. Also publicly acknowledge the methods of both students and tell the class it will be helpful to utilize both strategies for today’s task. During: While students work on the task, look for students (or groups) counting the horizontal and vertical travel time as well using the Pythagorean Theorem to make sense of each distance asked of them. Continue to ask questions like “How do you know it’s exactly twice as long?” or “What were your methods for finding this distance?”.

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Closure/Extension: It’s a good idea to start a conversation with the class about why they think the distance of persons A and B grow proportionally with their distance from the starting point. Students may start to come up with ideas and conjectures, but the goal here isn’t to come up with final conclusions. It’s just to produce a list of ideas to kick off the lessons to come.

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Walk it Out


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others.


Vocabulary: twice, path, distance, proportional, “distance traveled horizontally and vertically”

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

Students will work in groups and learn from the different strategies used.

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Lesson Series #2 Lesson Series Overview: The purpose of Lesson Series #2 is for students to get a sense of dilation through movement. Students will do activities where points start at the same location and then “move” a fixed amount of distance in different directions. Students should notice that the new distances between the two points are not only proportional to their distances from the starting point, but the segments representing each of their new distances are also parallel, which helps lead them to an understanding of AA. CCSS-M Standards Addressed: G.SRT.2, G.SRT.3 Time: 4 days


Description of Lesson: Students will dilate three collinear points, and see that the dilated points are also collinear, and that the two lines are parallel. They can show they are parallel by noticing that the slopes of the lines are equal.

Let’s All Walk it Out


Description of Lesson: Students verify whether potential “Walk it Out” results are possible. Students will need to use distances measured along the grid or with rulers to verify that the sides of the triangles are proportional.

Was that Even Possible?


Description of Lesson: Students will dilate three points not on a grid. Students will need rulers to complete this task, and may have misconceptions around which measurements they will need to take, and how to dilate a given point using these measurements. They should notice (by measuring) that the distances of the third pairs or the corresponding sides (if looked at as a triangle) are doubling, as a result of our doubling the first two pairs of sides.

Similarity Transformation

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Description of Lesson: Students will show that knowing two pairs of corresponding angles are congruent is enough to show that the triangles are similar, and that there exists a set of transformations (including dilation, if the shapes are different sizes), that carries one onto the other. Students will then do practice problems that will involve deciding whether there is a set of transformations (including dilations) that carries a given shape onto another, which establishes that they are similar. Notes: As a team, along with advisors from the district, we decided that because a dilation gives similar shapes, it is enough to show that if two shapes have the properties of similar shapes, they are the result of a dilation. This did not seem thorough enough to us to show that AA is a similarity shortcut, but the proof was agreed to be too advanced for the average secondary classroom. You can find the proof on the Illustrative Math website if you are interested in tackling it: http://www.illustrativemathematics.org/illustrations/1422 In a deductive Euclidean system, however, AA is also a postulate and not a theorem.

Angle-Angle Similarity Shortcut

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Expert Task Proving Proportionality



Math Objectives: ● This task will help students to use what they’ve learned about dilations

and similarity to justify conclusions about proportionality. ● Students will be given an opportunity to demonstrate their

understanding of similarity/dilation/proportionality through the use of a two-column proof.

CCSS-M Standards Addressed: G.SRT.4 Potential Misconceptions:

● Students may not recognize that the nested angle is the angle by which both triangles share.

● Students may have a difficult time seeing the nested triangles as two separate triangles. They may see a triangle and a trapezoid instead.

Launch: Students should start with a Do Now that helps them to recall all that they’ve been learning in terms of similarity and dilation. A possible problem could have something to do with any of the following topics: ● Parallel lines ● Parallel line angle relationships ● Dilations ● Similarity ● Proportionality

During: After introducing the task to teams, it would be great to give each team some time to create a rough draft or sketch of how they plan on proving the proportional relationship. This rough draft will help them create the two-column proof based off of what they know about the given diagram. As teams begin to form their two-column proof, you may choose to assign a poster component to this task to share each team’s thinking about how to prove the proportionality. It is important that students are given enough time to complete this task to construct an accurate and thoughtful proof. Closure/Extension: As teams finish sharing their thinking with the rest of the class, there can be a follow-up to this task in giving students a similar drawing and asking them to prove proportionality, if possible. This can take the form of an exit-ticket or pop-quiz. A possible question could be:

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Proving Proportionality


Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others. 6. Attend to precision.


Vocabulary:


Groups

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Lesson Series #3 Lesson Series Overview: The purpose of Lesson Series #3 is to help students identify and use geometrical knowledge to solve a problem. More specifically, the purpose of this lesson series is to help students:

● Make a mathematical model of a geometrical situation. ● Draw diagrams to help with solving a problem. ● Identify similar triangles and use their properties to solve problems. ● Track and review strategic decisions when solving a problem.

CCSS-M Standards Addressed: G.SRT.5 Time: 3 days


Description of Lesson: This is an introduction to the Formative Assessment Lesson called “Floodlights,” which gives students a problem to solve that requires their use of modeling similar triangles. Day 1 consists of discussions about shadows in general, as well as an individual assessment task. Students are encouraged to just access any ideas they already know and maybe uncover misconceptions before jumping into the larger task (in Day 2). Notes: See the notes in the Resource Folder in order to anticipate possible common issues and suggested questions/prompts (see T-4).

FAL: “Floodlights” T-1 through T-4 Introduction/Overview Common Issues


Description of Lesson: Once students have thought about shadows and the initial Floodlights task, then this lesson is an opportunity for students to collaborate in their thinking about how to more precisely model the situation. Students will be using sample responses (provided in the FAL resources) to review and analyze in their thinking. Notes: At the end of the lesson, be sure to give students the “How Did You Work?” (see P-5) questionnaire to gather further assessment information that will be helpful in planning Day 3.

FAL: “Floodlights” T-5 through T-10 Suggested Lesson Outline Solutions

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Description of Lesson: Day 3 is intended to be a follow-up lesson to the bulk of work completed in Day 2. Using the questionnaire that students completed the previous day (or for Homework if you wish), students could receive their individual solutions back and then compare their work to one another as well as to the sample responses that were provided in the Resources. This lesson can also be spent asking students to produce a fresh, complete and correct solution to the problem using one of the methods discussed in the previous lesson. Notes: Using posters for team’s to present their collective solution may be a good idea for this lesson so that students have to review/collaborate on all that they’ve learned about solving a problem involving similar triangles. Complete FAL is included in the Resources folder.

FAL: “Floodlights” T-5 through T-10 Suggested Lesson Outline Solutions

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Milestone Task Lessons from Abroad



Math Objectives: ● Students will use similar triangles to solve a real-world problem. ● Students will identify whether or not triangles are similar and explain

why. CCSS-M Standards Addressed: G.SRT.2, G.SRT.5 Potential Misconceptions:

● Students could also potentially get stuck on finding the length of BF in the diagram if they don’t know to use the Pythagorean Theorem or the distance formula.

Launch: Give students a warm-up problem involving similar triangles. During: Have graph paper available in case students want to make a scale drawing. Closure/Extension: Have students create their own real-world problems involving similar triangles.

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Lessons from Abroad


Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively.


Vocabulary: similar


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