Grade 9 – Module 4 Module Focus Session

Grade 9 – Module 4 Module Focus Session
February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Powerpoint Projector Clicker Laptop Document camera Participant Binder A Story of Functions A Close Look at Grade 10 Module 2

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: none Participant Poll Classroom teacher Math trainer Principal or school leader District representative / leader Other 2 min. In order for us to better address your individual needs, it is helpful to know a little bit about you collectively. Pick one of these categories that you most identify with. As we go through these, feel free to look around the room and identify other folks in your same role that you may want to exchange ideas with over lunch or at breaks. By a show of hands who in the room is a classroom teacher? Math trainer? Principal or school-level leader District-level leader? And who among you feel like none of these categories really fit for you. (Perhaps ask a few of these folks what their role is). Regardless of your role, what you all have in common is the need to understand this curriculum well enough to make good decisions about implementing it. A good part of that will happen through experiencing pieces of this curriculum and then hearing the commentary that comes from the classroom teachers and others in the group.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: none Session Objectives Examination of: The development of mathematical understanding across the module using a focus on concept development within the lessons. Examples that highlight themes and changes according to the Common Core State Standards. 2 min. Our objectives for this session are to: Examination of the development of mathematical understanding across the module using a focus on Concept Development within the lessons. Examples that demonstrate themes and changes according to the Common Core State Standards.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Agenda Orientation to Materials (if needed) Review of Module 1: Congruence, Proof, and Constructions Themes and Examples Module 2: Similarity, Proof, and Trigonometry 1 min. Here is our agenda for the day. If needed, we will start with orienting ourselves to what the materials consist of. Overall, I’d like to spend our session discussing the overarching themes of Modules 1 and 2. The idea is to leave with an understanding of where the major shifts in the Geometry are and use examples to make sense of those changes. (Click to advance animation.) Let’s begin with an orientation to the materials for those that are new to the materials (Skip if participants are already familiar with the materials).

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None What’s In a Module? Teacher Materials Module Overview Topic Overviews Daily Lessons Assessments Student Materials Daily Lessons with Problem Sets Copy Ready Materials Exit Tickets (Not accounted for in the timing – these slides are optional if participants are new to the materials.) Each module will be delivered in 3 main files per module. The teacher materials, the student materials and a pack of copy ready materials. Teacher materials include a module overview, and topic overviews, along with daily lessons and a mid- and end-of-module assessment. (Note that shorter modules of 20 days or less do not include a mid-module assessment.) Student materials are simply a package of daily lessons. Each daily lesson includes any materials the student needs for the classroom exercises and examples as well as a problem set that the teacher can select from for homework assignments. The copy ready materials are a single file that one can easily pull from to make the necessary copies for the day of items like exit tickets that wouldn’t be fitting to give the students ahead of time, as well as the assessments.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: None Types of Lessons Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept. Socratic Teacher leads students in a conversation to develop a specific concept or proof. Exploration Independent or small group work on a challenging problem followed by debrief to clarify, expand or develop math knowledge. Modeling Students practice all or part of the modeling cycle with real-world or mathematical problems that are ill-defined. (Not accounted for in the timing – these slides are optional if participants are new to the materials.) There are 4 general types of lessons in the 6-12 curriculum. There is no set formula for how many of each lesson type we included, we always use whichever type we feel is most appropriate for the content of the lesson. The types are merely a way of communicating to the teacher, what to expect from this lesson – nothing more. There are not rules or restrictions about what we put in a lesson based on the types, we’re just communicating a basic idea about the structure of the lesson. Problem Set Lesson – Teacher and students work through a sequence of 4 to 7 examples and exercises to develop or reinforce a concept. Mostly teacher directed. Students work on exercises individually or in pairs in short time periods. The majority of time is spent alternating between the teacher working through examples with the students and the students completing exercises. Exploration Lesson – Students are given 20 – 30 minutes to work independently or in small groups on one or more exploratory challenges followed by a debrief. This is typically a challenging problem or question that requires students to collaborate (in pairs or groups) but can be done individually. The lesson would normally conclude with a class discussion on the problem to draw conclusions and consolidate understandings. Socratic Lesson – Teacher leads students in a conversation with the aim of developing a specific concept or proof. This lesson type is useful when conveying ideas that students cannot learn/discover on their own. The teacher asks guiding questions to make their point and engage students. Modeling Cycle Lesson --Students are involved in practicing all or part of the modeling cycle (see p. 62 of the CCLS, or 72 of the CCSSM). The problem students are working on is either a real-world or mathematical problem that could be described as an ill-defined task, that is, students will have to make some assumptions and document those assumptions as they work on the problem. Students are likely to work in groups on these types of problems, but teachers may want students to work for a period of time individually before collaborating with others.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: None What’s In a Lesson? Teacher Materials Lessons Student Outcomes and Lesson Notes (in select lessons) Classwork General directions and guidance, including timing guidance Bulleted discussion points with expected student responses Student classwork with solutions (boxed) Exit Ticket with Solutions Problem Set with Solutions Student Materials Problem Set (Not accounted for in the timing – these slides are optional if participants are new to the materials.) Follow along with a lesson from the materials in your packet. The teacher materials of each lesson all begin with the designation of the lesson type, lesson name, and then 1 or more student outcomes. Lesson notes are provided when appropriate, just after the student outcomes. Classwork includes general guidance for leading students through the various examples, exercises, or explorations of the day, along with important discussion questions, each of which are designated by a solid square bullet. Anticipated student responses are included when relevant – these responses are below the questions; they use an empty square bullet and are italicized. Snapshots of the student materials are provided throughout the lesson along with solutions or expected responses. The snap shots appear in a box and are bold in font. Most lessons include a closing of some kind – typically a short discussion. Virtually every lesson includes a lesson ticket and a problem set. What you won’t see is a standard associated with each lesson. Standards are identified at the topic level, and often times are covered in more than one topic or even more than one module… the curriculum is designed to make coherent connections between standards, rather than following the notion that the standards are a checklist of items to cover. Student materials for each lesson are broken into two sections, the classwork, which allows space for the student to work right there in the materials, and the problem set which does not include space – those are intended to be done on a separate sheet so they can be turned in. Some lessons also include a lesson summary that may serve to remind students of a definition or concept from the lesson.

Key Changes in CCSS that Impact M1 & M2
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Key Changes in CCSS that Impact M1 & M2 “The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes generally)… In the approach taken here, two geometric figures are defined to be congruent if there is a sequence of rigid motions that carries onto the other.” 4 mins Module 1, focused on Congruence, (and Module 2, focused on Similarity) are where some of the most significant changes are occurring in geometry vs. traditional curriculum. This is due to the way congruence and similarity are defined under the CCSS, which is with the use of ‘geometric transformation’ as stated in this quote. Traditionally, we have called two segments for example congruent if they had the same lengths. It is not untrue to make this statement, however the CCSS lens allows us to declare figures as congruent in one fell swoop as opposed to making many individual measurements. Another benefit is that this approach allows us to broaden the kinds of figures we are able to compare at all. A ‘complex’ figure with not just straight edges but also curves is now among the kinds of figures we can compare. Since rigid motions preserve distances and angles, we have a way to try and manipulate the figure onto another figure for comparison. Before, by using the lengths of segments as the means of identifying two figures as congruent, we were really limited in the types of figures we could compare (we could really only compare rectilinear figures). Common Core State Standards for Mathematics, p74

Key Changes in CCSS that Impact M1 & M2
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Key Changes in CCSS that Impact M1 & M2 Transformations How they are first introduced The manner in which they are described and studied Their use in the definition of congruence and similarity Other uses to which transformations are put, e.g., reasoning and steps in proofs 2 mins So, very broadly, what is meant by ‘geometric transformation’? Are ‘transformations’ under the CCSS the same as they were in Regents Geometry? Well, yes and no. Yes, because the four transformations studied in this course are question are translations, reflections, rotations, and dilations; no because the treatment they get under the CCSS is different. So how they are introduced, the way in which they are studied, what we use them towards is different. Just to remind ourselves of what has traditionally been evaluated on transformations, here are a few recent Regents questions pertaining to transformations.

January 2014 Geometry Regents
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute January 2014 Geometry Regents 1 min

January 2014 Geometry Regents
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute January 2014 Geometry Regents 1 min These are two recent questions, but we could look through more and see that 1) transformations have been associated with the coordinate plane and 2) that there are a set of formulas that govern how transformations behave.

A Checklist of “rules”:
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute A Checklist of “rules”: 4 min

Module 1: Congruence, Proof, and Constructions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Module 1: Congruence, Proof, and Constructions Topic A: Basic Constructions Topic B: Unknown Angles Topic C: Transformations/Rigid Motions Topic D: Congruence Topic E: Proving Properties of Geometric Figures Topic F: Advance Constructions Topic G: Axiomatic Systems 2 mins Module 1 is a 45 day module, in fact it is one of two 45-day modules (Module 2, Similarity is the other). The bulk of class time is spent on Topics A-D, so let’s talk a little about the story that weaves these topics together. Clearly, Topic C is a focus as it is going to inform how we define what congruence is. So what are Topics A and B for? Well, the purpose of looking at constructions and unknown angles is to develop the basic language of geometry. In other words, we need to know WHAT we will be transforming. It wouldn’t make sense to talk about transformations before you know what you’re transforming or before you have a way of talking about what you’re transforming.

Topic A: Basic Constructions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic A: Basic Constructions Standards G-CO.1, G-CO.12, G-CO.13 Constructions How to: construct an equilateral triangle copy an angle bisect an angle construct a perpendicular bisector determine the incenter and circumcenter of a triangle 4 mins All the constructions in Topic A should look familiar (they have been part of the NY state standards and were tested on the Geometry Regents). What is different here is that the focus is not solely about the procedure or the “how to”, but being about to communicate the “how to” (how to tell people to ‘how to’). The ability to describe a CONstruction precisely is the ability to give an INstruction. In this study of constructions we are extending the meaning of vocabulary and seeing it in action. We are practicing communicating ideas to one another. Why do this? Why not just stick to the procedure? If we step back and think of what it means to understand the content deeply, we could say that Euclid represents the end of the spectrum of what it means to understand geometry deeply. He took all his understanding of geometry, started with a set of assumptions and using a deductive approach, built fact upon fact, and communicated the ideas effectively and efficiently in the books that make up Elements. Students are on the spectrum. And to work their way along the spectrum, they need to practice communicating these mathematical ideas and give them meaning beyond the initial level of just “doing” the basic action, e.g. a construction. Students do just this right off the bat in Lesson 1.

Lesson 1: Construct an Equilateral Triangle
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 1: Construct an Equilateral Triangle Flow of the lesson: Opening Exercise: (1) A question that jogs students ideas on distance (2) key vocabulary review Example 1: Guide students to discovering how to construct an equilateral triangle (with the use of a compass and straightedge). Example 2: Read an excerpt from Elements that describes Euclid’s approach to constructing an equilateral triangle; have students develop their own set of steps on how to construct an equilateral. Geometry Assumptions*: Introduce the notion that geometry is an axiomatic system- everything in the subject can be traced back to a short list of basic assumptions we take for granted. 1 min We are going to take a look at Lesson 1, and you will have an opportunity to experience it as students will experience it. The general flow of the lesson is as follows: [refer to the screen and the four major parts of the lesson]

Lesson 1: Construct an Equilateral Triangle
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 1: Construct an Equilateral Triangle 4 mins -- Allow participants 30 seconds to read/think about the Opening Exercise prompt, and another 30 seconds to exchange ideas. Encourage them to consider what students might say. Share out 1-2 responses. -- Allow participants 30 seconds to read/think about the vocab terms, another few seconds to exchange answers, and finally, share out responses. -- Have a poster ready of circle notation. Instruct participants that they may want to have such a poster ready, or some allocated space in their room to show this notation before reaching Example 2. A circle with center C and radius AB is written as CircleC, radius CB. segment radius circle

February 2014 Network Team Institute Lesson 1, Example 1 2 mins -- For a 2.5 hour slot, the idea is to introduce participants to what the flow of the lesson is like, but not to do these individual pieces of the lesson. Example 2 is where this curriculum really differs from traditional curriculum, so we are simply showcasing what is going on up until Example 2. In Example 1, you will lead students to helping them discover how to use their compass and straightedges to determine the location of the third cat.

February 2014 Network Team Institute Lesson 1, Example 1 1 min -- Show solution.

February 2014 Network Team Institute Lesson 1, Example 2 2 mins -- Refer to the Euclid excerpt, Proposition 1. Tell participants that they can have students annotate the text and use it as a guide to writing what they believe is a clear set of instructions on how to construct an equilateral triangle. Tell them to consider using the points from Example 1 as the points to refer to in the steps. -- Depending on who is in the audience, consider asking someone who feels they know the construction to describe the appropriate steps to the construction (maybe flip back to the solution image of the location of Mack). Elicit an informal set of steps.

February 2014 Network Team Institute Lesson 1, Example 2 4 mins -- Refer to the Euclid excerpt, Proposition 1. Tell participants that they can have students annotate the text and use it as a guide to writing what they believe is a clear set of instructions on how to construct an equilateral triangle. Tell them to consider using the points from Example 1 as the points to refer to in the steps. -- We want to highlight Example 2, as it is a prime example of students communicating their understanding about why the construction works vs. strictly completing the procedure. To recap up until this point- students have been primed to think about distance between points, have reviewed key vocabulary, have explored how to perform the construction at hand, and now working on shaping their thoughts and language to articulate how to perform the construction. This last part emphasizes deeper understanding, especially as students compare and try out each others’ steps to constructions, and inherently involves the Math Practices (MP 3, MP 6). Throughout Topic A, students will practice not only the constructions, but paying careful attention to the instructions that determine the construction. We want students to understand that a construction may not ever be completely perfect, due to human error, but the instructions that describe how to perform a construction do describe a perfect construction.

Topic B: Unknown Angles
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic B: Unknown Angles Standards G-CO.9 Breakdown of Lessons 6-11: Lessons 6-8: Unknown angle problems involving angles and lines at a point, transversals, and angles in a triangle. Students use simple justification in non-algebraic steps. Lessons 9-11: Unknown angle proofs, non-numeric problems emphasizing justification of a relationship, sometimes a known existing relationship. 2 mins The motivation in Topic B is similar to that in Topic A: there is an actual task and then a step towards being able to discuss or explain the task. With unknown angle problems, students are actually solving a problem but also beginning to use justifications from their knowledge of facts related to angles. With unknown angle proofs, numbers are replaced by variables, and the focus on geometric relationships is sharper, and requires students to be attentive to the reasons that allow them to take the next step or draw the next conclusion in a problem. The practice with numeric problems makes for a smooth transition to reasoning without numbers. Work with angles, even single step unknown angle problems (i.e. solving for the missing angle), began in Grade 4. Students see an especially closer look at unknown angle problems in Grade 7, Module 3. It is worth noting here that as part of the transition from Topics A to B, an extensive chart of facts associated with angles learned since Grade 4 appears in the Problem Set of Lesson 5 (the last lesson in Topic A). This primes students with the review of needed facts to (1) solve problems and (2) explain why they take the algebraic steps they take in a problem.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic B: Unknown Angles Lesson 8, Exercise 2: Lesson 9, Discussion: 4 mins Remember that Lessons 6-8 are unknown angle problems, therefore the first example is about solving for the measure of a and providing a justification for how you arrived at that answer. -- Allow participants the opportunity to answer the question and cite a reason. -- Step through to show the second example. What do you notice about the second example relative to the first example? -- Elicit that it is non-numeric and that the goal here is to establish a relationship of some sort. You might want to say, well I still know that the sum of two remote interior angles is equal to the measure of the respective exterior angle. But is it possible to actually show this with the use of any other facts? What can be established about x, y, and z?

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic B: Unknown Angles What can be established about x, y, and z? 3 mins -- Take participant suggestions about establishing a relationship between x, y, and z before reviewing the solution. The unknown angle proofs require students to really break down relationships, including ones they know to be true; they call on simpler facts to explain newer ones. The process is deductive. w

Lesson 10, Unknown Angle Proofs- Proofs with Constructions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 10, Unknown Angle Proofs- Proofs with Constructions Problem Set, 1 AB DE and Prove that m ABC = m DEF. BC EF 4 mins -- This is an additional problem, something to highlight another Unknown Angle Proof lesson. Depending on time, step participants through the problem and solution.

Topic C: Transformations/Rigid Motions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic C: Transformations/Rigid Motions Standards G-CO.2, G-CO.3, G-CO.4, G-CO.5, G-CO.6, G-CO.7 G-CO.12 Key concepts in Lessons 12-21: Formal definition of each rigid motion (rotation, reflection, translation) and how each rigid motion is related to constructions Compositions of rigid motions Applications of rigid motions: discussion regarding correspondence, how congruence is defined 2 mins We have laid the ground work in Topics A and B: we have reviewed both precise vocabulary and called explicit attention to facts that we have been learning over several years, and now we are ready to tackle transformations, specifically rigid motions. If Topics A and B represent the “what” in geometry, figures we study, extending their meaning by using them in constructions, then in Topic C, transformations tell us how things relate or compare to one another, they help us establish what the relationship is between two things. We eventually use this for a very particular relationship, one where the rigid motions help establish whether two things are the same or not. Specifically, students will learn the formal definitions of each rigid motion, and how to apply them to figures, how each transformation is tied to constructions, what compositions of rigid motions look like, and toward the end of Topic C, we explain congruence in terms of transformations.

February 2014 Network Team Institute Rigid Motions in Grade 8 What knowledge do students arrive with? Rigid motions first introduced in Grade 8, Module 2 G8 focus: Understanding of what a transformation is Intuitive understanding and experimental verification of the properties of the rigid motions (translation, reflection, and rotation). Descriptive definitions Hands-on exploration with the use of transparencies 1 min Students study rigid motions in Grade 8, chiefly in a hands-on and descriptive approach to build intuitive understanding of how the rigid motions behave.

February 2014 Network Team Institute Grade 8, Module 2, Lesson 4 Grade 8 introduction to reflections: 3 mins In grade 8 (M2, L4), reflections are introduced in the following hands-on way. Students have a piece of paper with a point Q on line L represented in black ink -- Summarize the Grade 8 experience, the following doesn’t need to be said verbatim: - Let L be a vertical line and let P and A be two points not on L as shown below. Also, let Q be a point on L. (The black rectangle indicates the border of the paper.) The following is a description of how the reflection moves the points P, Q, and A by making use of the transparency: - Trace the line L and three points onto the transparency exactly, using red. (Be sure to use a transparency that is the same size as the paper.) - Keeping the paper fixed, flip the transparency across the vertical line (interchanging left and right) while keeping the vertical line and point Q on top of their black images. - The position of the red figures on the transparency now represents the reflection of the original figure. Reflection(P) is the point represented by the red dot to the left of L, Reflection (A) is the red dot to the right of L, and point Reflection(Q) is point Q itself. Note that point Q is unchanged by the reflection. The red rectangle in the picture on the next page represents the border of the transparency. In the picture above, you see that the reflected image of the points is noted similar to how we represented translated images in Lesson 2. That is, the reflected point P is P'. More importantly, note that the line L and point Q have reflected images in exactly the same location as the original, hence ReflectionL=L and Reflection(Q)=Q, respectively. Pictorially, reflection moves all of the points in the plane by “reflecting” them across L as if L were a mirror. The line L is called the line of reflection. A reflection across line L may also be noted as Reflection(L).

(Grade 10) Lesson 14: Reflections
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute (Grade 10) Lesson 14: Reflections Exploratory Challenge 1 min Lesson 14 begins with a tie to constructions. Students discover that the line of reflection coincides with the perpendicular bisector of each segment determined by a pair of corresponding vertices in the figure. For segments AA’, BB’, and CC, the perpendicular bisector of each coincides with the line of reflection DE. Recall that students have studied the construction of the perpendicular bisector in Lesson 4 (see next slide for reference).

Reflections and the Perpendicular Bisector Construction
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Reflections and the Perpendicular Bisector Construction 2 mins In fact, students first learned the construction for an angle bisector, and how observations on the verification of an angle bisector construction led to the perpendicular bisector construction. Students verified an angle bisector construction by folding along the angle bisector– a look at the segment EG demonstrated whether the construction was done correctly. When the angle is folded along the bisector AJ, E should coincide with G, in fact a correct construction showed that E was as far from F as G was, or F was the midpoint of EG. Furthermore, by folding along the bisector of a correctly done construction meant that angles EFJ and GFJ should coincide, have the same measure, and since they lie on a straight line, should each have a measure of 90˚. With this understanding, and with the knowledge of the steps of the angle angle bisector construction, students determine steps of the perpendicular bisector and have an understanding of the implications of a perpendicular bisector.

February 2014 Network Team Institute Lesson 14: Reflections Example 1 2 mins They use this understanding to determine a line of reflection between two figures in Example 1. Based on the Exploratory Challenge, how is the line of reflection determined? -- Allow the audience an attempt at the response. Students must join any two corresponding points between the figures and cosntruct the perpendicular bisector in order to determine the line of reflection.

February 2014 Network Team Institute Lesson 14: Reflections Example 1 1 min This solution shows the perpendicular bisector of CC’ constructed; any two of the corresponding points used would be acceptable and would have yielded the same results.

February 2014 Network Team Institute Lesson 14: Reflections Definition of Reflection: 2 mins These exercises lead to the formal definition of reflection. Just to review-- this definition comes after an intuitive understanding is developed and student have had experience manipulating reflections and testing out their properties with transparencies (G8), and after making a connection with constructions from the beginning of Module 1. Students have an understanding of how reflections behave and know, by construction, why they work the way they work.

February 2014 Network Team Institute Lesson 14: Reflections Definition of Reflection: 1 min In addition to however you want to break the definition down with students, the lesson provides an examination of each part of the definition.

February 2014 Network Team Institute Lesson 14: Reflections Application of Reflection: 2 mins These are exit ticket questions for this lesson. Students are able to (1) determine the line of reflection between a figure and it’s reflected image and (2) reflect a figure across a line of reflection. Each of the three transformation lessons builds to the formal definition and then works on the application of the transformation to figures. Each one also makes use of constructions. For example, in studying rotations, students discover that the center of rotation between a figure and its rotated image can be found with the help of perpendicular bisectors. Applying a translation to a figure involves the construction of parallel lines. So we see the progression of the topics beginning to come together.

Lesson 19: Construct and Apply a Sequence of Rigid Motions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 19: Construct and Apply a Sequence of Rigid Motions We define two figures in the plane as congruent if there exists a finite composition of basic rigid motions that maps one figure onto the other. The idea of “Same size, same shape” only paints a mental picture; not specific enough. It is also not enough to say that two figures are alike in all respects except position in the plane. A congruence gives rise to a correspondence. 3 mins Congruence is defined in L19 after a study of each basic rigid motion, as well as a look at symmetry (as a rigid motion) in Lesson 15, and the Parallel Postulate in Lesson 18. - The idea of “Same size, same shape” only paints a mental picture; not specific enough: Just as it is not enough to say, “Hey he looks like a sneaky, bad guy who deserves to be in jail”, it is not enough to say that two figures are congruent if they have the same size and same shape- it lacks specificity needed in a mathematical argument. - It is also not enough to say that two figures are alike in all respects except position in the plane. In defining congruence as a finite composition of basic rigid motions that maps one figure onto another, we need to be able to refer and describe this rigid motion. Additionally, a congruence by one rigid motion and a congruence by a different rigid motions are two separate things. Specifying one of many possible rigid motions may be important. - A congruence gives rise to a correspondence. A correspondence between two figures is a function from the parts of one figure to the parts of the other, with no conditions concerning same measure or existence of rigid motions. In other words, two figures do not have to be congruent for a correspondence to exist, but a congruence always yields a correspondence. If there exists a rigid motion T that takes one figure to another, then a natural correspondence results between the parts. For example, if a figure contains a segment AB, then a congruent figure includes a corresponding segment T(A)T(B).

February 2014 Network Team Institute Topic D: Congruence Standards G-CO. 7, G-CO.8 Key theme in Topic D: 3 mins In Grade 7, students discovered that deciding whether two triangles are identical to each other does not mean that 6 measurements (3 side lengths and 3 angle measures) of both triangles had to be known. For example, they discovered that if only 3 side lengths of each triangle were known, they could predict whether the triangles would be identical or not (if the triangles had the same side length measures, they would indeed be identical because there was only one way those lengths could be put together to form a triangle; the three sides condition). However in Grade 10, the goal is to explicitly prove why these “shortcuts” work; prior to the CCSS, we used these patterns to forecast that if the parts in the shortcut (e.g. A-S-A) were the same for both triangles, then all three side lengths and all three angle measures were the same for both triangles. Even if we know this information, we are now defining figures to be congruent if a rigid motions maps one figure to another. Students will see that the shortcuts are indicators that a rigid motion does exist in the cases those indicators hold true, but there are ways to describe the actual sequence of rigid motions that maps one figure onto a another figure.

Lesson 22: Congruence Criteria for Triangles- SAS
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Congruence Criteria for Triangles- SAS 1 min. We are going to show that the SAS criteria for triangles indicates that the two triangles ABC and A’B’C’ are congruent because there exists a sequence of rigid motions that will map one triangle onto the other. The two triangles are distinct. We are trying to determine if one triangle will map to the other, or coincide with the other. If the goal is to see what happens when they coincide, we must use a transformation that will bring them together.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Congruence Criteria for Triangles- SAS 1 min A translation by vector A’A will result with the two triangles coinciding at A. Could we have translated by say B’B or C’C? No, because we are working with parts we know to be equal in measure, and angle A and angle A’ are of equal measure.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Congruence Criteria for Triangles- SAS 1 min Continuing in our goal to try and make the triangles coincide, we can use a rotation of d degrees about A to bring AC’ to AC. We know C’ will map to C because by the assumption AC and AC’ are the same length and a rotation is distance preserving, so there is no change in any length in the transformation.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Congruence Criteria for Triangles- SAS 5 mins Again, in the pursuit to see if the triangles coincide, we see B and B’’’ are on opposite sides of AC. We use a reflection over the line that contains AC so that B’’’ maps to B. How do we know that B’’’ definitely maps to B? Because rigid motions angle measures, we know angle B’’’AC=BAC and therefore ray AB’’’ maps to AB. Additionally, by assumption we know that AB=AB’’’, so between these two facts, B’’’ definitely maps to B. This sequence of rigid motions takes triangle A’B’C’ to triangle ABC with the use of just three known criteria from each triangle: two side lengths and the included angle measure. We can generalize this argument for any two triangles with the same criteria (distinct or coinciding along some part of the triangle). Hence we have proven that when the SAS criteria is satisfied between two triangles, there always exists a rigid motion that will map the one triangle onto the other. Students will complete proofs for all the triangle congruence criteria using rigid motions, as well as the otherwise accepted fact that base angles of an isosceles triangle are equal in measure.

Lesson 26: Triangle Congruency Proofs – Part I
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 26: Triangle Congruency Proofs – Part I Exercise 5 1 min Once the the congruence criteria are proved (and any known fact is proved), we are free to call on them in problems.

Topic G: Axiomatic Systems
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Topic G: Axiomatic Systems Standards All A mathematician’s story: Many years ago I taught calculus for business majors. I started my class with an offer: "We have to cover several chapters from the textbook and there are approximately forty formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas." The students gladly agreed. 3 mins Topic G is a summary of definitions, geometric assumptions, geometric facts, constructions, unknown angle problems and proofs, transformations, and proofs that establish properties we previously took for granted. All of the mathematicians involved in this curriculum wanted to be sure that we remember that geometry, like other subjects in math and science is a logical, deductive system. That the “beauty” of this subject (and in other deductive systems) is that the glue of what holds the subject together, is the process of examining what you know, and examining, critically, why you know it. This is what Euclid did in writing Elements, and the text has been honored for over 2000 years because of the how meticulously he put it together. Having said this, it is not part of this curriculum to understand geometry from the ground up, but to recognize that there is a short list of facts from which all other facts are derived. An in-depth study of the assumptions is left for college-level classes. The 8 geometric assumptions, or axioms, are seen in a handful of lessons (namely 4 lessons: L1, L3, L12, and L18). We see four of them mentioned in Lesson 1. The assumptions are highlights in the lesson; they are not meant to be -- Anecdote to share: Many years ago I taught calculus for business majors. I started my class with an offer: "We have to cover several chapters from the textbook and there are approximately forty formulas. I may offer you a deal: you will learn just four formulas and I will teach you how to get the rest out of these formulas." The students gladly agreed.

Example from Mid-Module Assessment
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Example from Mid-Module Assessment 5 mins This is an example from the Mid-Module Assessment (that also pertains to what has been discussed in this presentation). I know we did not have an in-depth look at rigid motions as the students will have once they arrive at this question, but please take a few moments and formulate a response. -- Allow participants 2 minutes to consider and discuss amongst themselves.

Example from Mid-Module Assessment
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Example from Mid-Module Assessment 3 mins -- Review solutions.

Key Themes of Module 1: Congruence, Proof, and Constructions
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Key Themes of Module 1: Congruence, Proof, and Constructions Module is anchored by the definition of congruence Emphasis is placed on extending the meaning and use of vocabulary in constructions There is an explicit recall and application of facts learned over the last few years in unknown angle problems and proofs Triangle congruence criteria are indicators that a rigid motion exists that maps one triangle to another; each criterion can be proven to be true with the use of rigid motions. 2 mins That wraps up the portion of our session on Module 1. -- Review all four bullet points; these are the key themes in the module.

Module 2: Similarity, Proof, and Trigonometry
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Module 2: Similarity, Proof, and Trigonometry Topic A: Scale Drawings Topic B: Dilations Topic C: Similarity and Dilations Topic D: Applying Similarity to Right Triangles Topic E: Trigonometry 1 min Module 2 is the other 45 day module in the curriculum. It too is a place where the course sees a significant shift versus traditional curriculum, this time because of the way similar is defined under the CCSS. Just as congruence is defined through the lens of transformations, so is similarity. In addition to the need for rigid motions, similarity requires the discussion of another transformation, dilation.

Topic A: Scale Drawings
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic A: Scale Drawings Standards G-SRT.1, G-SRT.4 Key Concepts Two ways two construct scale drawings: Ratio Method and Parallel Method Triangle Side Splitter Theorem Dilation Theorem 2 mins Topic A, Scale Drawings, is a natural place to begin the discussion that leads to dilations, as they are a concept familiar to students (G7, M1). They know a scale drawing of a figure is a magnification or reduction of a figure, and it has side lengths that are in constant proportion to the corresponding side lengths of the original figure, and have angle measures that are equal to the corresponding angle measures of the original figure. Work in grade 7 is more heavily observational and students verify these properties by calculations and measurements. Here, students are creating scale drawings. They two formal methods of creating scale drawings. In the last two lessons of Topic A, the use the two different methods to draw scale drawings yields two theorems that will be used repeatedly to prove the properties of dilations in Topic B.

February 2014 Network Team Institute Lesson 2: Ratio Method 2 mins To use the Ratio Method requires an understanding of how to dilate points, which is first addressed in Grade 8, Module 3. -- Review definition of dilation.

Lesson 2: Ratio Method, Example 1
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 2: Ratio Method, Example 1 2 mins By the Ratio Method of drawing scale factors, key vertices are dilated about a center O and scale factor r. A ray is drawn from the center of dilation through each vertex. Then, according to the scale factor, an appropriate distance is measured and marked as the dilated vertex. Once key vertices are located under the dilation, the dilated points are joined.

February 2014 Network Team Institute Lesson 2, Example 2 5 mins Use the handout and ruler to scale the following figure (Lesson 2, Example 2) according to the Ratio Method. -- Allow participants time to create the scale drawing. Step 1. Draw a ray beginning at O through each vertex of the figure . Step 2. Use your ruler to determine the location of A' on OA ; A' should be 3 times as far from O, as A. Determine the locations of B' and C' in the same way along the respective rays. Step 3. Draw the corresponding line segments, e.g., segment A'B' corresponds to segment AB.

Lesson 3: Parallel Method, Example 2
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 3: Parallel Method, Example 2 2 mins By the Parallel Method of drawing scale factors, an initial dilated vertex is provided or must be located (in the same way as by the way points are determined by the Ratio Method). Then a set square and ruler are used to draw segments parallel to the segments of the initial figure using the initial point. Students first use set squares in Grade 7; there is a refresher exercise at the beginning of this lesson (L3).

February 2014 Network Team Institute Lesson 3, Example 2 5 mins Use the handout and ruler to scale the following figure (Lesson 3, Example 1) according to the Parallel Method. Use the figure below, center O, a scale factor of r = 2 and the following steps to create a scale drawing using the parallel method. Step 1. Draw a ray beginning at O through each vertex of the figure. Step 2. Select one vertex of the scale drawing to locate; we have selected A'. Locate A' on ray OA so that OA'=2OA. Step 3. Align the setsquare and ruler as in the image below; one leg of the setsquare should line up with side AB and the perpendicular leg should be flush against the ruler . Step 4. Slide the setsquare along the ruler until the edge of the setsquare passes through A'. Then, along the perpendicular leg of the setsquare, draw the segment through A' that is parallel to AB until it intersects with OB; label this point B'.

Consequences of the Ratio and Parallel Methods
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Consequences of the Ratio and Parallel Methods Students prove that the Ratio and Parallel Methods produce the same scale drawing for a give figure Resulting theorems used frequently in Topic B: Triangle Side Splitter Theorem. A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side. Dilation Theorem. If a dilation with center O and scale factor r sends point P to P’ and Q to Q’, then P’Q’ = r PQ . Furthermore, if r≠1, and O, P, and Q are the vertices of a triangle then PQ P’Q’. 2 mins Once students are familiar with the Ratio and Parallel Methods of creating scale drawings, they establish that both methods in fact create the same scale drawing of a given figure. Proving this yields the Triangle Side Splitter Theorem, which in turn is used in the following lesson to establish the Dilation Theorem. The proof for the Dilation Theorem explains why it is that the the Ratio and Parallel Methods yield scaled or enlarged/reduced versions of a given figure. These two theorems are used over and over again to establish properties of dilations in Topic B lessons.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic B: Dilations Standards G-SRT.1, G-SRT.4 Key Concepts Proving properties of dilations, including how it is that dilations map segments to segments, lines to lines, rays to rays, and angles to angles. 1 min Topic B explores dilations inside and out, from the definition to the explicit proof of why the properties of dilations (such as why it is that a dilation of a segment maps to another segment).

Lesson 7: How Do Dilations Map Segments?
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 7: How Do Dilations Map Segments? Example 3 To show that dilations map segments to segments, call on: The definition of a dilation The Triangle Side Splitter Theorem Construction of a ray For r < 1 4 mins Students prove that a dilation maps a segment to another segment (in this case a segment, but the proof is generalized later for a line). This proof is an example of the step by step reasoning students have to apply to reach a conclusion. They use the definition of a dilation to establish that the dilated segment P’Q’ splits the sides PQ and OQ proportionally, after which they call on the Triangle Side Splitter Theorem to establish that the lines that contain PQ and P’Q’ are parallel. Then they have to try and establish why it is that all the points between P and Q map to all the points between P’ and Q’. Students are provided time to come up with their own ideas before being provided a “model” proof, which makes use of a construction of a ray drawn from O so that it intersects PQ at R, and consequently it intersects P’Q’ at R’ . Since R' belongs to P'Q' by construction, and we already know that P'Q' is parallel to PQ, then P'R' must be parallel to PR. This is an instance where the Triangle Side Splitter Theorem is called again on to show that P’R’ splits triangle OPR proportionally. Finally the definition of a dilation is called on a second time to show that the arbitrary point R is sent to R’; we can use this reasoning to account for every point between P and Q being sent to a point between P’ and Q’. The argument is a great example of MP 1, 3, and 7. Similar arguments are used for

Topic C: Similarity and Dilations
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic C: Similarity and Dilations Standards G-SRT.2, G-SRT.3, G-SRT.4, G-SRT.5 Key Concepts Similarity Transformation. The composition of a finite number of dilations and/or rigid motions. Two figures are said to be similar if a similarity transformation exists maps one figure onto another. A congruence is a similarity with scale factor 1. 2 mins With a thorough understanding of dilations, we then define a similarity transformation in Topic C as the composition of a finite number of dilations and/or rigid motions, and we say two figures are similar if there exists a similarity transformation that maps one figure onto another. Just as the use of rigid motions allowed us to compare figures other than rectilinear figures, so too does the idea of a similarity transformation allow us to compare and determine whether any two figures are similar to each other.

February 2014 Network Team Institute Lesson 12: What are Similarity Transformations and Why Do We Need Them? Example 1 Describe a similarity transformation that maps figure Z onto figure Z’. 4 mins Traditionally, when you open to the similarity chapter of a textbook, it quickly focuses on triangles as the chief shape to study, so much so that again, like congruence, when you ask students about what they remember regarding congruence and similarity, they may be able to recite the ‘shortcuts’ (SSS, ASA, SAS for similarity, etc), but the understanding ends with the abbreviations. You’ll notice that the first lessons in Topic C use curvilinear images to drive home the point that we can compare any kind of figure with the aid of similarity transformations. Having said that, there is still much to be said about triangles and similarity, and the topic addresses the criteria that identifies two figures as similar (i.e. AA, SAS, and SSS criteria). Step 1. The dilation will have a scale factor of r<1 since Z' is smaller than Z. Step 2: Notice that Z' is flipped from Z1. So take a reflection of Z1 to get Z2 over a line l. Step 3: Take a translation that takes a point of Z2to a corresponding point in Z'. Call the new figure Z3. Step 4: Rotate until Z3 coincides with Z'.

Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Discussion 5 mins The AA criterion for similar traingles is covered in Lesson 15, and the SAS and SSS criteria for similar triangles is covered in Lesson 17. Students prove that a similarity transformation exists for two triangles with two pairs of angles of equal measure. Take a moment to sketch a rough outline of what this proof might look like. Hint: Use a dilation with center A (r < 1…but you can be specific about this)….you will need triangle congruence indicators as well. Theorem. Two triangles with two pairs of equal corresponding angles are similar.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar 4 mins In a nutshell: Dilate about A with r < 1…in fact specifically so that r = DE/AB , so that B goes to B’ and C goes to C. Since we have dilated B and C by the same scale factor, B’C’ is a proportional side splitter; by the Triangle Side Splitter Theorem, we know that B'C'||BC. Since B'C'||BC, then m∠AB'C'=m∠ABC because corresponding angles of parallel lines are equal in measure. Then △AB'C'≅△DEF by ASA. Thus, a similarity transformation exists that takes △ABC to triangle △DEF; triangle △ABC is similar to △DEF. Since the triangles are similar, we can confirm that the Angle Angle criterion between two triangles guarantees that the triangles are similar. Once the proof is established, students can use what they know about the length relationships between similar triangles to solve for unknown sides, much like problems found in current text in similarity units. Theorem. Two triangles with two pairs of equal corresponding angles are similar.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Exercise 9 3 mins The AA criterion for similar traingles is covered in Lesson 15, and the SAS and SSS criteria for similar triangles is covered in Lesson 17. Students prove that a similarity transformation exists for two triangles with two pairs of angles of equal measure. In a nutshell, they will show that a dilation of a triangle about one vertex can create a triangle congruent to the partner triangle. Then students can call on a triangle congruence indicator to establish that a transformation exists that will map one triangle to the other. Once the proof is established, students can use what they know about the length relationships between similar triangles to solve for unknown sides, much like problems found in current text in similarity units.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar Problem Set 4 2 mins Take a moment to attempt this question. -- Allow a few moments for participants to attempt the question.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 15: Angle-Angle Criterion for Two Triangles to be Similar 2 mins

Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar 4 mins The SAS and SSS criteria for two triangles to be similar is covered in Lesson 17. Take a moment to sketch a rough outline of what this proof of why the SAS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = A'C/'AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Angle-Side Congruence Criterion.

Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 17: SAS and SSS Criteria for Two Triangles to be Similar 3 mins Take a moment to sketch a rough outline of what this proof of why the SSS criterion is enough to determine that two triangles are similar. Hint: Use a dilation about A. -- Allow participants to consider for a few moments. The proof of this theorem is simply to take any dilation with scale factor r = A'B’/AB = B'C’/BC = A'C’/AC. This dilation maps △ABC to a triangle that is congruent to △A'B'C' by the Side-Side-Side Congruence Criterion.

Lessons 19 and 20: Modeling Lessons using Similarity
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 19 and 20: Modeling Lessons using Similarity Eratosthenes’ calculation of the Earth’s circumference 1 min A fitting use of similarity is seen in the ancient Greek, Eratosthenes’, calculation of the circumference of the Earth. With an impressive use of observation, measurement, and understanding of similarity, he calculated a close approximation of the Earth’s circumference. Roughly around the same time (approximately 250 BC), another ancient Greek named Aristarchus approximated the distance from the Earth to the moon. Both of these calculations are presented to students in Lessons 19 and 20. Aristarchus’ calculation of the distance from the Earth to the Moon

Topic D: Applying Similarity to Right Triangles
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic D: Applying Similarity to Right Triangles Standards G-SRT.6, G-SRT.7, G-SRT.8 Key Concepts The special relationships that result when an altitude is drawn in a right triangle from the vertex of the right angle to the hypotenuse. Proving the Pythagorean Theorem using similarity Simplifying radical expressions 2 mins With an understanding of what a similarity transformation is, what it means for two figures to be similar, and an understanding of the triangle similarity criteria, students then study similarity within the scope of right triangles. The major concepts studied are the similarity relationships that arise when an altitude is drawn from the vertex of a right triangle to the hypotenuse, how to use similarity to prove the Pythagorean Theorem, and how to simplify radical expressions, specifically multiplying and dividing radical expressions and adding and subtracting radical expressions using the Distributive Property.

February 2014 Network Team Institute Lesson 21: Special Relationships with Right Triangles- Dividing into Two Similar Sub-Triangles Discussion We can use the AA criterion to show that each of the smaller two right triangles is similar to the large right triangle; similarity is transitive, so all three triangles are similar. 3 mins Students understand that the altitude of a right triangle from the vertex of the right angle to the hypotenuse divides the triangle into two similar right triangles that are also similar to the original right triangle. They initially study this in Grade, Module 3 (?) but revisit it here as a way to help prepare for trigonometry. How can we show that △ABC~△BDC~△ADB? -- Allow the audience a minute to discuss among themselves? The altitude of a right triangle splits the triangle into two right triangles, each of which shares a common acute angle with the original triangle. By the AA criterion, △ABC and △BDC are similar and △ABC and △ADB are similar and since similarity is transitive we can conclude that △ABC~△BDC~△ADB.

February 2014 Network Team Institute Lesson 21: Special Relationships with Right Triangles- Dividing into Two Similar Sub-Triangles 1 min We want to emphasize the use of the equal values of corresponding ratios to solve for the unknown values. Eventually this focus will be shifted to how the ratio is dependent on a given acute angle of the right triangle. We highlight the importance of this concept by having students write out ratios of lengths within each triangle, for each of the three triangles.

February 2014 Network Team Institute Lesson 21: Special Relationships with Right Triangles- Dividing into Two Similar Sub-Triangles 4 min Take a moment and find the appropriate ratios for each table and triangle. -- Allow time to complete the tables.

February 2014 Network Team Institute Lesson 21: Special Relationships with Right Triangles- Dividing into Two Similar Sub-Triangles 2 min What would be one way to solve for x, using these ratios? x/5 = 5/13; x = 25/13 or 1 12/13 Why can we use this equation to solve for x in this way? Because corresponding ratios of sides are equal in value. And so we are beginning to steer students towards the idea of special ratios within right triangles and how they are a function of the acute angle within the right triangle, regardless of the magnitude of the triangle.

Lesson 22: Proving the Pythagorean Theorem by Similarity
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Proving the Pythagorean Theorem by Similarity Discussion How can the ratios in right triangles (e.g. shorter leg:hypotenuse) be used to prove the Pythagorean Theorem? 2 mins The use of the ratios of sides within right triangles continues in Lesson 22 as one way of proving the Pythagorean Theorem. Take a few moments to separately arrange and orient the three triangles within triangle ABC. -- Provide 2 minutes to complete this.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Proving the Pythagorean Theorem by Similarity Which right triangle ratios can be used to express a in terms of other variables? 3 mins In order to prove the Pythagorean Theorem, which ratios can be used to isolate and express a, b, or c in terms of other variables? -- Provide 2 minutes to complete this.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 22: Proving the Pythagorean Theorem by Similarity 2 mins We have three sets of ratios to choose from: shorter leg:hypotenuse, longer leg:hypotenuse or shorter leg:longer leg By using the longer leg:hypotenuse and shorter leg:hypotenuse ratios, we have a way to re-express a2 and b2, which by substitution is c2. longer leg: hypotenuse shorter leg: hypotenuse

February 2014 Network Team Institute Lessons 23 and 24: Multiplying and Dividing Expressions with Radicals Adding and Subtracting Expressions with Radicals 2 mins With this latest look at the Pythagorean Theorem, Lessons 23 and 24 focus on working with expressions with radicals. In grade 8 students learned the notation related to square roots and understood that the square root symbol automatically denotes the positive root (Grade 8 Module 7). In grade 9, students used both the positive and negative roots of a number to find the location of the roots of a quadratic function. We review what we know about roots learned in Grade 8, Module 7, Lesson 4, now because of the upcoming work with special triangles in this module. In Lesson 23, students multiply and divide expressions that contain radicals to simplify their answers. In Lesson 24, the work with radicals continues with adding and subtracting expressions with radicals.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: None Topic E: Trigonometry Standards G-SRT.6, G-SRT.7, G-SRT.8 Key Concepts Definition of sine, cosine, tangent Relationship between sine and cosine Application of the trig ratios to solve missing value, real world, and area problems. 1 min The final topic in Module 2 is Trigonometry. Here the basic definitions of sine, cosine, and tangent are addressed, as well as how they are applied in a variety of settings. Relative to changes in earlier in the module and to Module 1, the lessons on trigonometry will appear relatively familiar.

Lesson 25: Incredibly Useful Ratios
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 25: Incredibly Useful Ratios Quote from Lesson 25: “In fact if, since we know the values of these ratios are constant for a given angle measure, we could compile a table of every acute angle between 0˚ and 90˚ and list the values of the adj/hyp and opp/hyp ratios with respect to each acute angle measure.” 3 mins In this last lesson before defining the trig ratios sine, cosine, and tangent, we again highlight the ratios within right triangles. Students are now equipped with the language that describes a given side of a triangle with respect to a given acute angle in the right triangle. In the lesson, two groups take measurements of a given acute angle for a set of right triangles and the side lengths of the triangles. They calculate the adj/hyp ratio and the opp/hyp ratio. Eventually, they realize that the two groups have two sets of similar triangles once the angle measurements of the triangles are shared out loud. The quote above is suggested guidance for the teacher, who is slowly guiding them to see that the ratios that are constant in value between similar triangles is a function of the acute angle in question. Therefore, though the use of these ‘internal’ ratios is not new, the lesson shines light on the fact that ratios depend on the acute angle that the sides are labeled by and not the magnitude of the triangle. The explicit connection is made in Lesson 26 with the formal definition of the sine, cosine, and tangent ratios.

Lesson 27: Sine and Cosine of Complementary Angles and Special Angles
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 27: Sine and Cosine of Complementary Angles and Special Angles 1 min Students have observed a similar image in L26 with numbers, and predict the pattern there, but formalize what is actually happening in L27. The ratios for sin α and cos β are the same, so sin α=cos β and ratios for cos α and sin β are the same, so cos α=sin β. The sine of an angle is equal to the cosine of its complementary angle and the cosine of an angle is equal to the sine of its complementary angle. What conclusions can be drawn based on each of the ratios?

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 27: Sine and Cosine of Complementary Angles and Special Angles 2 mins Students are prefaced to the values of the ratios of the special angles with a discussion on how the values of sine and cosine change as theta changes between 0˚ and 90˚. As θ gets closer to 0, a decreases. Since sinθ=a/1, the value of sinθ is also approaching 0. b increases and becomes closer to 1. Since cosθ=b/1, the value of cosθ is approaching 1. As θ gets closer to 90, a increases and becomes closer to 1. Since sinθ=a/1, the value of sinθ is also approaching 1. b decreases and becomes closer to 0. Since cosθ=b/1, the value of cosθ is approaching 1.

Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lesson 27: Sine and Cosine of Complementary Angles and Special Angles 2 mins Students are introduced to the handful of special angle measures (and special triangles) that appear frequently in trigonometry. Students are presented with a table of the most common angle measures (0, 30, 45, 60, 90) and then use the following triangles to actually show how those values are found.

Lessons 28-29: Solving missing value problems
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Lessons 28-29: Solving missing value problems Selected examples to try: Lesson 28: Problems using Sine and Cosine, Exercise 6 Lesson 29: Applying Tangents, Example 2 5 min In Lessons students will use trig ratios to solve missing value type of problems. Please take some time to try the two selected examples from these lessons.

February 2014 Network Team Institute L28, Exercise 6 1 min -- Take responses from participants; review solution.

February 2014 Network Team Institute L29, Example 2 2 mins -- Take responses from participants; review solution.

Key Themes of Module 2: Similarity, Proof, and Trigonometry
Grade 9 – Module 4 Module Focus Session February 2014 Network Team Institute Key Themes of Module 2: Similarity, Proof, and Trigonometry Just as rigid motions are used to define congruence, so dilations are added to define similarity. A close look at the way dilations are constructed yields information about how dilations behave. Right triangle similarity is built on the concepts of within-triangle ratios and between-triangle ratios leading to an understanding of trigonmetric ratios. 2 min Lessons involve the application of the trig ratios, including how to use trigonometry to calculate area and the Laws of Sines and Cosines. -- Review key themes of M2.

February 2014 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes(175) MATERIALS NEEDED: Biggest Takeaway What is your biggest takeaway with respect to Modules 1 and 2? How can you support successful implementation at your school/s given your role? 3 mins Take a few minutes to reflect on this session. You can jot your thoughts on your copy of the powerpoint. What are your biggest takeaways? Now, consider specifically how you can support successful implementation of these materials at your schools given your role as a teacher, school leader, administrator or other representative.

Grade 9 – Module 4 Module Focus Session

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