A Story of Functions Grade 10-Module 1

A Story of Functions Grade 10-Module 1
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X A Story of Functions Grade 10-Module 1 Congruence, Proof, and Constructions 1 min Welcome and introduction.

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Session Objectives Articulate and model the instructional approaches to teaching the content of the first half of the lessons. Examine how the topics and lessons promote mastery of the focus standards and address the major work of the grade. Articulate connections from the content of previous grade levels to the content of this module. 2 mins Read aloud objectives. As we move through this session, please ask questions that will help with your immediate understanding of the material. If you have questions that relate to your broader understanding of A Story of ratios or how to implement the curriculum in your school or district, please write those on a sticky note along with your name and place the note on the parking lot. We will look at those questions during the lunch break and address them with the group if they are applicable to all or with you if they are specific to your situation. Principals in particular- You will want to make notes for yourself as a result of your observations to use in later sessions tomorrow. .We will be focusing you on the areas where you will want to make notes for yourself to use later.

Participant Poll Classroom teacher
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Participant Poll Classroom teacher School leader (Math AP, Department Chair, etc.) Principal District leader BOCES representative 3 mins In order for us to better address your individual needs, it is helpful to know a little bit about you collectively. Who of you are classroom teachers? (Call for a show of hands.) School-level leader? Principal? District-level leader? BOCES representative? NOTE TO FACILITATOR: As you poll the participants, take note of the approximate size of each group. This will make it easier for you to re-group the participants for the final portion of this presentation. Regardless of your role, what you all have in common is the need to deeply understand the mathematics of the curriculum and the intentional instructional sequence in which it is brought to life for students. Throughout this session, we ask you to be cognizant of your specific educational role and how you will be able to promote successful implementation in your classroom, school, district, and/or BOCES. Each time we pause to reflect, please do so through the lens of your own professional responsibilities. At the close of this session, you will have the opportunity to share your thoughts, ideas, and concerns with others in a similar role.

Agenda Module 1 Overview Constructions Expert Lesson Group Work
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Agenda Module 1 Overview Driving concepts Constructions Basic compass work Lesson 1 Expert Lesson Group Work Expert Lesson Presentations Group Walk Activity Summary of Work & Closure 2 mins Review the Agenda for the session.

Module 1: Congruence, Proof and Constructions
Header July 2013 Network Team Institute Module 1: Congruence, Proof and Constructions 1 min Tell participants that we are spending time today looking at the first of five modules. Modules 1 and 2 are the lengthiest in terms of days needed, at roughly 45 days each.

Header July 2013 Network Team Institute Icebreaker! Each table needs a poster paper and no more than two markers. A vocabulary word will be given. Once you see/hear it, write down as many words as possible related to the vocabulary word. The table with the most words wins! You have 2 minutes to work. Anyone can write, but only with the two markers provided. Ready, Set, Go! 5 mins Read through the instructions on the slide. Give tables 2 minutes to write as many words as possible related to the topic “angles.” At the end of the 2 minutes teams need to count the number of words they came up with. Have all participants stand. Select one table to say their number of words aloud. Then say “If you have less that ## then you can sit down.” Repeat until only one table is standing. Deem them the winners and provide a cheap prize (optional). ANGLES

Module 1 Overview Topics A-C and Lesson Titles Focus Standards
Header July 2013 Network Team Institute Module 1 Overview Topics A-C and Lesson Titles Focus Standards Foundational Standards Mathematical Practice Standards New and Familiar Terms 8 min. Say, “You may have an idea of what is in Module 1 if you attended a previous NTI meeting. This time, I’d really like you to give a thorough read of the module overview so you know exactly what to expect from the lessons.” Provide participants time to read through module overview. Think about the module overview through the lens of your role. How might you use this information in your role? ( It is a quick summary of what teachers will do for the next few weeks and can be shared with parents and the community, it also summarizes what conversations should be happening in grade level meetings and what resources teachers may be requesting for their classrooms, it can be shared with other teachers in the school for the purpose of thematic unit planning.)

Header July 2013 Network Team Institute Driving Concepts The module culminates in the concept of congruence and its application in proof problems. Two figure are congruent if there exists a finite composition of basic rigid motions (rotations, reflections, translations) that maps one figure onto the other figure. Discussion of congruence requires an understanding of rigid motions Proof problems require geometric justification (practiced through Unknown Angles) and an application of transformations Transformations are inherently linked to constructions, i.e., the concept of a perpendicular bisector is essential to reflections 5 mins The major concepts in Module 1 stem from the concept of congruence, as defined under the Common Core State Standards (read the definition for ‘congruent’ aloud). The topics are very much connected. To discuss congruence is to discuss rigid motions, hence, Topic D: Congruence is immediately preceded by Topic C: Transformations/Rigid Motions. Proofs, which dominate the end of the module, incorporate congruence, transformations (i.e. the proof of why the base angles of equal measure of a triangle imply that the triangle has two sides of equal length is neatly shown with the help of a reflection over the angle bisector of the vertex), and geometric justification. Students practice justifying each step they take of a problem in topic B: Unknown Angles. And finally, constructions and the careful use of language to perform constructions are the basis of transformations.

Working with Compasses
Header July 2013 Network Team Institute Working with Compasses What to expect when working with compasses: What kind of experience do students have with compasses? What kinds of difficulties should we anticipate from students? What problems should we anticipate regarding the materials themselves? 5 mins What to expect when working with compasses: What kind of experience do students have with compasses? Most likely little to none. What kinds of difficulties should we anticipate from students? Managing a compass, pencil sliding out, holes in the paper What problems should we anticipate regarding the materials themselves? Directed to administrators: Depending on the type of compass, might need a screwdriver to tighten them periodically, possibly need LOTS of golf pencils, readily available sharpeners for all those golf pencils, issues around plastic compasses breaking…etc.

Construction Essentials
Header July 2013 Network Team Institute Construction Essentials Studying constructions requires: Ability to use a compass An understanding of the importance of labeling How to follow and write instructions 1 min We would like to prime our brains for the first five lessons and consider what students need in order to be successful with the lessons.

Header July 2013 Network Team Institute Activity 1 Construct 3 circles of different sizes. 1. Use geometric vocabulary to describe what circles of “different sizes” mean? 5 mins 1. Use geometric vocabulary to describe what circles of “different sizes” mean? Circles of different sizes have radii of different measures.

Header July 2013 Network Team Institute Activity 2 Construct the figure described in the steps below. Compare your construction to that of a neighbor. Draw circle K. Label the endpoints of the diameter of circle K as A and B. Draw circle A with radius AK. Label endpoints of the diameter of circle A as KL. Draw circle B with radius BK. Label endpoints of the diameter of circle B as KM. If LA = x, what expression represents AM? 5 mins Take a few moments and follow the constructions steps provided. What does your figure look like? Compare your figure with someone else’s. Did you construct the same exact figure?

Activity 2 Construction
Header July 2013 Network Team Institute Activity 2 Construction Share out any interesting observations you notice about the figure. Can we do this problem WITHOUT labels? 8 mins Review the questions and ask the audience to share out their comments. Share out any interesting observations you notice about the figure. Outer circles pass through the center of the inner circle, all the endpoints of the diameters are collinear… Can we do this problem WITHOUT labels? The use of labels are essential to the completion of a construction; without labels we would be lost. 3. What difficulties might students face with this problem? 4. Consider this problem: imagine the same figure with no labels. Write the steps to perform the construction. How would students manage this task? 3. What difficulties might students face with this problem? 4. Consider this problem: imagine the same figure with no labels. Write the steps to perform the construction. How would students manage this task?

Constructions and Mathematics Practices
Header July 2013 Network Team Institute Constructions and Mathematics Practices Which Mathematical Practices are inherent to the successful completion of Topic A? MP 5 – Use appropriate tools strategically MP 6 – Attend to precision MP 7 – Look for and make use of structure 2 mins - Ask the participants which MPs they think go hand in hand with the constructions topics.

Header July 2013 Network Team Institute Lesson 1 Opening Exercise Joe and Marty are in the park playing catch. Tony joins them, and the boys want to stand so that the distance between any two of them is the same. Where do they stand? How do they figure this out precisely? What tool or tools could they use? 2 mins - Facilitator leads whole group through the SF of Lesson 1.

Header July 2013 Network Team Institute Lesson 1 Vocabulary 1 The _______ between points and is the set consisting of , , and all points on the line between and . 2 A segment from the center of a circle to a point on the circle. 3 Given a point in the plane and a number , the _______ with center and radius is the set of all points in the plane that are distance from the point . 3 mins - Review the terms segment, radius, and circle from SF.

Lesson 1 Example 1 Header July 2013 Network Team Institute 2 mins
This exercise is based on a YouTube video found by googling “equidistant cats”. Ask participants to try Example 1 to locate Mack. Discuss approaches and difficulties encountered, noting that 10th graders will struggle with where to begin. Note that this presents teachers with an opportunity to stress the need for students to persevere (MP 1).

Lesson 1 Header July 2013 Network Team Institute 2 mins
Review solutions

Lesson 1 Example 2 Header July 2013 Network Team Institute 10 mins
Review Proposition 1 with participants. The goal is to annotate the text and discover the concise way to instruct a reader to perform the construction. Say: What difficulties will students encounter with this? (“old-fashioned” language, unfamiliar terms) How did Euclid’s construction respond to the “equidistant cats” problem? How did Euclid’s approach differ from yours (or from what you expect your students will do)?

Lesson 1 Geometry Assumptions Header July 2013 Network Team Institute
- Highlight this section as a section that will appear a handful of times in Module 1. We revisit the assumptions at the very end of the module, when we discuss Geometry as an axiomatic system. Say: - One can create a completely logical statement from nonsensical assumptions. - Example: Given that all cows are purple (assumption), and that the creature in the field is a cow (precondition), then the creature is purple (conclusion). While the conclusion is absurd, it follows logically from the system’s assumptions and preconditions. - Fortunately, Euclid worked with more reasonable assumptions. Again, Euclid’s assumptions hold true under his logical system. Non-Euclidean geometry uses a different set of assumptions. Teachers will bring students to understand that the reason for proofs in geometry has little to do with stating that something fairly obvious is true (e.g. ABCD is a square) and much to do with creating logical, step-by-step irrefutable arguments under a given logical system – the whole reason Abraham Lincoln studied Euclid to become a better lawyer. - Tell participants that the “geomhistory” site is included to allow teachers leeway in presenting information. It falls under he “nice to know” instead of the “need to know” category.

Lesson 1 Relevant Vocabulary
Header July 2013 Network Team Institute Lesson 1 Relevant Vocabulary Geometric Construction. A geometric construction is a set of instructions for drawing points, lines, circles and figures in the plane. The two most basic types of instructions are: Given any two points and , a ruler can be used to draw the line or LAB segment (Abbreviation: Draw .) Given any two points and , use a compass to draw the circle that has center at and that passes through (Abbreviation: Draw circle: center , radius .) Constructions also include steps in which the points where lines or circles intersect are selected and labeled. (Abbreviation: Mark the point of intersection of the lines and by , etc..) 2 mins - Review undefined terms: point, line, plane, distance along a line, distance around a circular arc. - Students are led through a set of terms which are clearly defined and are told the appropriate notation to use when referring to specific geometric terms.

Header July 2013 Network Team Institute Lesson 1 Figure. A (2-dimensional) figure is a set of points in a plane. Equilateral Triangle. An equilateral triangle is a triangle with all sides of equal length. Collinear. Three or more points are collinear if there is a line containing all of the points; otherwise, the points are non-collinear. Length of a Segment. The length of the segment is the distance from to , and is denoted or . Thus, Coordinate System on a Line. Given a line , a coordinate system on is a correspondence between the points on the line and the real numbers such that (i) to every point on there corresponds exactly one real number, (ii) to every real number there corresponds exactly one point of , and (iii) the distance between two distinct points on is equal to the absolute value of the difference of the corresponding numbers. 3 mins - After the term length of a segment, students are led through a discussion of notation regarding lengths of segments to expose them to the various notations they will encounter throughout their reading in geometry. - Students will be reminded of the need for precision in language as they describe steps in constructions and/or state their reasoning in proofs. Knowing the meanings of the terms is a critical prerequisite for describing constructions and formulating proofs.

Header July 2013 Network Team Institute Lesson 1 Exit Ticket We saw two different scenarios where used the construction of an equilateral triangle to help determine a needed location (i.e., the friends playing catch in the park, and the sitting cats). Can you think of another scenario where the construction of an equilateral might be useful? 3 mins Ask participants for ideas. (e.g., location of fire hydrants or sprinklers)

Header July 2013 Network Team Institute Lesson 1 Problem Set 1. Write a clear set of steps for the construction that determines the final location of Margie’s cat, Mack. Use Euclid’s Proposition 1 as a guide. 7 mins Ask: - What constructions will be needed? -- two circles, three line segments - Which terms will students use in their steps? – circle, center, radius, intersection - Why is there only one possible location for the cat? – the room is too small to include the other point of intersection

Header July 2013 Network Team Institute Lesson 1 Problem Set 2. Suppose two circles are constructed using the following instructions: Draw circle: Center , radius . Under what conditions (in terms of distances , , ) do the circles have i) One point in common? ii) No points in common? iii) Two points in common? iv) More than two points in common? Why? 5 mins Ask participants to work this problem out, including the constructions of the circles. Discuss the results. Notice the TF discussion about using the lengths of the segments to justify responses. See next slide.

Lesson 1 Problem Set Header July 2013 Network Team Institute 2 mins
- Walk through this with participants. Ask: - Were you able to find all four constructions and explain them using segment lengths?

Header July 2013 Network Team Institute Lesson 1 Problem Set 3. You will need: A compass and straightedge Cedar City boasts two city parks and is in the process of designing a third. The planning committee would like all three parks to be equidistant from one another to better serve the community. A sketch of the city appears below, with the centers of the existing parks labeled as P1 and P2. Identify two possible locations for the third park and label them as P3a and P3b on the map. Clearly and precisely list the mathematical steps used to determine each of the two potential locations. 5 mins Ask: How does this problem relate to the lesson? How is it similar to Margie’s cat problem? How is it different?

Lesson 1 Problem Set Header July 2013 Network Team Institute 2 mins
Ask: Are there now two possible locations? How should students justify their choice of the two? Take one minute to reflect on this session. How do these lessons compare to your past experiences with mathematics instruction? What are the implications for the supports and resources your colleagues will need to fully implement this curriculum with fidelity? Jot down your thoughts. Then you will have time to share your thoughts. Give participants 1 minute for silent, independent reflection. Turn and talk with a partner at your table about your reflections. Allow 2 minutes for participants to turn and talk about their reflections. Then, facilitate a discussion that leads into the key points on the next slide

Expert Lesson Group Work
Header July 2013 Network Team Institute Expert Lesson Group Work Read through the rest of the lessons in Topic A (Lessons 2-5) to get an overall view of the lesson content. Your group will be responsible for leading the discussion for one specific problem from one of the these four lessons (randomly assigned). Become an Expert! You and your group will be presenting a 5 minute mini-lesson to the whole group on how to solve your assigned problem. Complete the task in 35 min. 35 mins Now we will dig deeper into the lessons themselves. What you are going to do is become an expert on one problem from one lesson from Topic A in this module. To become an expert you will need to do the following:” Read the first 4 bullet points on the slide. Say, “At the end of the work session you and your table mates will need to present a complete solution to your group’s assigned problem. You will have 35 minutes to become an expert and prepare your presentation. Are there any questions about the task?” Hand out the problems, one per group. Say, “Go ahead and begin.” Circulate while tables work so that you can answer questions and clarify the task. (The specific tasks are: Lesson 2 Example 1 and Exit Ticket; Lesson 3 Problem Set # 2 and #5; Lesson 4 Construct perpendicular bisector and Construct Perpendicular to a Line through a Point; Lesson 5 Construct a Circumcenter and Construct a Circumcenter.)

Expert Lesson Group Work
Header July 2013 Network Team Institute Expert Lesson Group Work

Header July 2013 Network Team Institute Topic A Basic Constructions (G-CO.1, G-CO.12, G-CO.13) Lesson 1: Construct an Equilateral Triangle Lesson 2: Construct an Equilateral Triangle II Lesson 3: Copy and Bisect an Angle Lesson 4: Construct a Perpendicular Bisector Lesson 5: Points of Concurrencies 1 mins - Now that you have constructed equilateral triangles, copied and bisected angles, constructed perpendiculars and perpendicular bisectors, and determined points of concurrencies in triangles, you are ready to move to the world of unknown angles. - Students will revisit and refine their earlier understandings of angles and lines at a point, of transversals and of triangles. The construction work in Topic A and the deepening knowledge of angle congruence in these first few lessons of Topic B will allow students to begin writing precise informal and formal proofs.

Prep for Lessons 6-8: Unknown Angles
Header July 2013 Network Team Institute Prep for Lessons 6-8: Unknown Angles Three types of problems: Angles and Lines at a Point Transversals Angles in a Triangle 5 min Say, “The Problem Set for Lesson 5 is a review of facts about these three types of problems that students have previously encountered in earlier grades. Briefly review these facts with your group.” - Allow groups 5 minutes to read/discuss the Lesson 5 Problem Set from the TF, then ask if there are any questions.

Unknown Angles Group Walk Activity
Header July 2013 Network Team Institute Unknown Angles Group Walk Activity Eight problems are posted around the room. At the signal, your group will move to one of the problems and solve it. One group member should record the solution. When the signal is given, rotate to the next problem. After your group has solved all problems, return to your table. 15 mins - Review activity procedures from the slide. - Hand out an answer sheet to each group. - Allow about two minutes for each problem before having groups rotate. - Discuss the activity. Ask: “Did any one problem cause more difficulty than the others? How well did your group work together to arrive at solutions?

Header July 2013 Network Team Institute Topic B Unknown Angles (G-CO.9) Lesson 6: Angles and Lines at a Point Lesson 7: Transversals Lesson 8: Angles in a Triangle 1 min - We have now worked with two of the three foundational tools for proofs in geometry – constructions and angle relationships. Tomorrow, we will dive into proofs beginning with the familiar territory of unknown angles. We will then lay the final foundation for proofs of congruence – transformations.

Biggest Takeaway Driving Concepts
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Biggest Takeaway Driving Concepts The module culminates in the concept of congruence and its application in proof problems. Discussion of congruence requires an understanding of rigid motions Proof problems require geometric justification (practiced through Unknown Angles) and an application of transformations Transformations are inherently linked to constructions, i.e., the concept of a perpendicular bisector is essential to reflections 3 mins - Early in this presentation, we talked about the driving concepts in Module 1 as outlined above. - Our whole goal is to drive home to students the concept of congruence and its application in proof problems. To get this accomplished, we need to get students to understand rigid motions and to formulate precise proofs. Transformations are inherently linked to constructions. Therefore, we are laying the foundation for proofs of congruence by having students work with constructions and by solving unknown angle problems.

Key Points Precision in construction and in language is critical.
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Key Points Precision in construction and in language is critical. Perpendicular bisectors are a key component in two of the three rigid motions – reflections and rotations. Logical systems require a clearly articulated set of assumptions upon which to build. 2 mins Take questions. Now that we’ve examined all aspects of the module, let’s consider your plans for training, implementation, and differentiation. Take a minute and tie all that we have together and think about it from the lensed of the role you fill. Teacher- time to read through the modules, discuss the lessons with a co-teacher, solve the problems, find/order materials, highlight key questions School Leader- identify areas that may need the most support, consider how to bridge the gaps in student understanding, identify what teachers need to know to effectively implement the modules, find/order materials Principal- provide time for teachers to plan, facilitate discussions around modules to encourage team problem solving, consider how the module impacts the way teachers demonstrate lesson planning, consider how this impacts what will been seen during an observation, organize/support the process of gathering/ordering materials District Leader – provide funding and resources necessary for staff development, curricular materials, and assessment. Consider implications of new curriculum and new tests and prepare stakeholders for change. BOCES Representative – Identify economies of scale for staff training. Consider implications of new curriculum and new tests and prepare stakeholders for change. Turn and talk with a partner at your table about your reflections. What, for you, is the biggest takeaway? Jot down your thoughts. Allow 2 minutes for participants to turn and talk about their reflections. Then, facilitate a brief discussion that leads into the key points on the next slide.

Day 1: Afternoon Session

Agenda Take the Mid-Module Assessment Table Discussion
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Agenda Take the Mid-Module Assessment Table Discussion Rubric Scoring with Student Exemplars Summary & Closure 1 min Read through agenda.

Mid-Module Assessment
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Mid-Module Assessment Take the Mid-Module Assessment 20 minutes No talking, group work, etc.. 2 min. Explain the process. Say, “We will examine the mid-module assessment today. First you will take the assessment. Spend 20 minutes working quietly and individually. Go ahead and begin working.”

Mid-Module Assessment Discussion
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Mid-Module Assessment Discussion Table Discussion Predict the errors that students will make Identify vocabulary or context that students may struggle with Discuss strategies to overcome these issues that will support student success Read the instructions on the slide for the discussion piece. Give 10 minutes for tables to discuss the assessment.

Mid-Module Assessment Scoring
Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Mid-Module Assessment Scoring Rubric Scoring Each table has been provided a set of student exemplars Use the rubric to score the assessment After you have scored at least two assessments, compare the scores you gave with someone else. Discuss any discrepancies. Read the instructions on the slide for the scoring piece. Distribute the student exemplars. Give 40 minutes for tables to score and discuss the assessment and rubric scoring experience. For administrators, please think of the implications of using rubrics on your current grading policies, what are teachers going to need to discuss and agree upon when using rubrics versus their current grading structures? Consider the implications of this data (quantitative and qualitative) for the professional development needs of the staff, the resources needed to support struggling students, the conversations held in the data analysis meetings. You will want to make notes on ideas you have as a result of your observations to use in later sessions tomorrow.

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: X minutes MATERIALS NEEDED: X Summary and Closure What did you think about the scoring process in general? Final comments. Read the first bullet and provide tables 2-3 minutes to discuss the process in general. Ask for volunteers to share with the whole group.

A Story of Functions Grade 10-Module 1

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