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**A Story of Ratios Grade 7 Module 1 – First Half of Lessons**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Copies of Power Point A Story of Ratios Grade 7 Module 1 – First Half of Lessons Facilitator introductions and general welcome. (2 minutes) NOTE—distribute copies of the PowerPoint handout at this time. Say: During this morning’s session you will be actively engaged in unpacking the content of the first half of the lessons in module 1. You will be asked to interact with the materials from both the student’s and teacher’s perspective at various times during the session to deeply understand the content module one contributes to the story of ratios. As you gain familiarity with these materials, feel free to jot down thoughts and questions for discussion during our reflection times. So let’s jump right in for a busy three hours. (1 minute) 3 total minutes

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X 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. Read through bulleted items. Bullet 1: We want you be comfortable with how Module 1 content is going to be delivered, specifically the differences between what you’ve seen/experienced before and what is expected now. Bullet 2: We want you to understand how we have designed lessons to promote mastery of the standards and the major work of the grade. Bullet 3: Finally we will look at the coherence, or general upward spiral of the curriculum- how this module builds upon grade 6 study of ratios and rates providing consistency as students move through the grades 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.

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**Participant Poll Classroom teacher School leader Principal**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Participant Poll Classroom teacher School leader Principal District leader BOCES representative 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.

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**Agenda Review of Module Overview**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Agenda Review of Module Overview In-Depth Examination of Module 1 Topics A and B: Lessons 1-10 Analysis of Topic Openers Features of Student and Teacher Materials Modeling of Lesson Components Coherence Across Grade Levels Closure and Reflections 1 min Say: We’ll start with reviewing the module structure, then examine the Module Overview and Topic Openers. An examination of the Assessments will happen during the afternoon sessions. Next, we’ll study a lesson in great detail, uncovering the intentionality behind the instructional choices. In the time we have, it is not possible to do every lesson in its entirety so we have chose to highlight key components of each lesson, in the hopes of illustrating the arc of the module. So some parts of lessons we will do together, or give you time to examine or discuss. Other portions can be done later today or when you go home in more detail. As we examine key components from the lessons, we will take also make note how specific pieces or terminology has a role in the coherence across both the module as well as other grade levels. Finally, we will have time for closure and reflection. Let’s start our review of the module structure, which is consistent across all modules of all grades in A Story of Ratios, by taking a quick look at the curriculum map.

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**A Story of Ratios Curriculum Overview**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X A Story of Ratios Curriculum Overview 1 min Say: We will only be looking at module 1 for grade 7 in this session. Today will focus on the first half of lessons.

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**Review of Module Structure**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Review of Module Structure Module Overview Topic L1 L2 L3-L4 L5-L6 L7 L8-L9 L10 L11-L12 L13 L14 L15 L16 L17 L18 L19 L20 L21-L22 2 minutes Say: Let’s take a minute to review the organizational structure of A Story of Ratios: A Story of Ratios: A Curriculum Overview for Grades 6-8 provides a curriculum map and grade-level overview. The curriculum map provides an at-a-glance view of the entire story, making clear the coherence of the curriculum and the role that each module plays in that progression. 6-7 mods per grade Mods comprised of topics (number varies per mod) Topics vary in # of lessons Lesson designed for 45 minute instructional period; Modules are comprised of topics, topics break into concepts, and concepts become lessons. Graphic shows breakdown: Each component, moving from the Overview to the Lesson, provides a more specific level of information. As you plan to implement A Story of Ratios, each of these components will be important to your understanding of the instructional path of the module. The Standards, both Content and Practice, come to life through the lessons. Rigorous problems are embedded throughout the module. We will spend time in the sessions today and tomorrow exploring this further.

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**Grade 7 Module 1 Overview TIME ALLOTTED FOR THIS SLIDE: 3 minutes**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Module 1 overview in binder Grade 7 Module 1 Overview 3 min Turn to the Module Overview- pages 6 to 7. Give me a thumbs up when you find it. Say: For those of you either here in May or familiar with the materials presented during that time, we hope you recall a big idea from this module (as well as other modules in our curriculum) is the inclusion of precise mathematical language in our materials. Today we will look at how that is addressed within the lessons. As we were creating the lessons, we felt it was necessary to make some adjustments to the terminology and representations section of this Module Overview. Say: Can anyone here find something on these 2 pages that looks different from what you previously have seen? Look for participants to note the new/revised terminology: proportional to, revised symbol for constant of prop to k, and equation as y = kx, added one-to-one correspondence and used more detailed language for Scale drawing and scale factor. Say: We made these changes to be sure the language we are using in grade 7 is consistent with what students will see across grade levels, specifically setting up work with similar figures in grade 8 and geometry in high school Say: As we proceed through key lesson components today and tomorrow, we will discuss how these terms will be used with students. As a teacher, when I first looked at some of the revised terms, I was concerned about how students were going to be able to digest these descriptions. So, as our team designed lessons, we used them more for students to get at the concept behind each one instead of having students get bogged down with reciting the description. Say: 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.) *see next slide for representations*

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**Grade 7 Module 1 Overview TIME ALLOTTED FOR THIS SLIDE: 1 minute**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: X Grade 7 Module 1 Overview 1 min Say: As was discussed in the previous page, the equation change is noted here as well. Both y= cx and y = kx are used in the progressions document however we decided to use y = kx to limit any confusion for students as the progressions document also uses c in a different context as it refers to determining whether two ratios are equivalent (bottom of p.13). Say: Let’s next take a look at where this terminology and the representations will be addressed within the module as we look at what exactly is in Module 1. <<advance>> * Next slide gets into content by topic: “What’s in G7M1? “

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 6 minutes MATERIALS NEEDED: Module Overview What’s in G7-M1? Topic A: Explore what it means to be proportional to and how to determine if two types of quantities are in a proportional relationship, examining both tables and graphs Topic B: Define the constant of proportionality and use it to represent proportional relationships with an equation, interpret the meaning of key points on the graph- (0,0) and (1,r) where r is the unit rate Topic C: Compute unit rates involving fractions, find equivalent ratios of two partial quantities given a part-to-part ratio and the total of the quantities, solve multi-step ratio problems including markup, markdown, and commission Topic D: Explore scale drawings and recognize the term scale factor as the constant of proportionality 6 min SAY: Let’s quickly summarize the content of Module 1. Can someone paraphrase what topic A is about? (pause for 30 sec) (Allow for 1 audience member to say and then click to advance first bullet.) In topic A students explore what it means to be proportional, and whether two types of quantities are in a proportional relationship. As we shall experience in a moment, their exploration begins with a collaborative work task which also helps to set the tone for the year in creating strong classroom procedures at the same time as they are doing math from the first module. The experience ties together their study of ratios and rate from grade 6 and sets up the new study of proportional relationships in grade 7. What is topic B about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 2nd bullet.) In Topic B, students define the constant of proportionality, which they have already been exploring in Topic A, and use it to represent proportional relationships with an equation. They also take a close look at the graphs of proportional relationships and develop meaning of special points. What is topic C about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 3rd bullet.) In Topic C, they will apply their work with unit rates in grade 6 to work with fractional values given within the context of given situations. What is topic D about? (pause for 30 sec, allow for 1 audience member to say and then click to advance 4th bullet.) In this topic students will apply their understanding of proportional relationships as they explore scale drawings, and create a scale drawing of their own. Students will use discover that the scale factor is just another application of the constant of proportionality in a proportional relationship.

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**Topic Openers Read the concept chart and the descriptive narrative.**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Topic A Opener Topic Openers Read the concept chart and the descriptive narrative. Make note of the important information that will help teachers implement these lessons. 5 total min Turn to __ to find the topic opener for Topic A. Say: A topic opener like this is included at the start of each topic. We write these prior to designing the lessons. Each lesson is designed for a 45-minute class period. In some cases, a lesson will extend through 2 or 3 days, as noted here with Lessons 3-4 This is the topic opener for Topic A. Take one minute to read the topic opener including the concept chart, and then I’ll ask you to share your observations. While you are reading identify 3 pieces of information contained herein. Pause for 1 minute for participants to read. Share with your table what 3 pieces of information you found to be key. (Encourage participants to share their observations at table for 2 min., then share out for 2 min. Before moving on, make sure the following points are addressed, even if you need to state them directly. ) The focus standards (7.RP.2 and 7.RP.2a) are bold. The language of the focus standard, as it appears in the CCLS, is also provided. The number of instructional days (6) is provided. Lesson Titles and their types are referenced in parenthesis after each lesson- we’ll come back to this in a moment (Lesson Structure Key: P-Problem Set Lesson, M-Modeling Cycle Lesson, E-Exploration Lesson, S-Socratic Lesson) A narrative provides explanation as to how the standards are addressed in this Topic. The language here is more specific, providing lesson by lesson description to explain how lessons build on each other over the course of the module. They are designed to help build teacher capacity and knowledge.

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**Types of Lessons Problem Set Socratic Exploration Modeling**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Types of Lessons Problem Set Students and teachers work through examples and complete exercises to develop or reinforce a concept or procedure. 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 and teacher practice part of the modeling cycle with problems that are ill-defined and have a real world context. 5 min This slide will teach participants about the four types of lessons. Say: Number off at your tables 1, 2, 3, 4, 1, 2, 3, 4, etc.. If you are a 1 you will read out #1 out loud and then the next number 1 will summarize it with your table. The 2’s will read number 2 and the next number 2 will summarize it, etc.. For tables with more than 8 people, the extra persons can add any additional thoughts or clarifications on any of the lesson types. Before you begin reading or summarizing when it is your turn, briefly introduce yourself to your table partners. Pause for 3-4 minutes while participants discuss the types of lessons at their tables. Say: Does anyone have any questions? (pause and answer or move on) Say: Module 1 for this grade only contains Problem Set and Exploration lessons. Socratic lessons will only make up about 10% of our entire curriculum. Now that we have discussed the Topic Opener, let’s take a look at lesson components and how we will make this all come together for 7th grade students. NOTE TO FACILITATOR: See this background knowledge to help you address any questions. 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 with the goal of clarifying, expanding upon or developing a concept, definition, theorem or proof. Typically a challenging problem or question that requires students to collaborate (in pairs or groups) but can be done individually. Class discussion on the problem for a period of time (10 minutes might be appropriate) to draw conclusions and consolidate understandings. Socratic Lesson – Teacher leads students in a conversation with the aim of developing a specific concept or proof. Only about 10% of lessons fall under this category minutes devoted to student/teacher conversation. Useful when conveying ideas that students cannot learn/discover on their own. The teacher asks guiding questions to make their point and engage students. The remaining time could include a fluency activity to open the lesson or there may be a debrief or application problem at the end of the lesson. Modeling Cycle Lesson –At this level, students are involved in practicing part of the modeling cycle. The problem students are working on is ill-defined and has a real world context. 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 their group members. Students and teacher work through the modeling cycle in a reduced form to complete an application problem over 1 or 2 days.

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**Lesson 1: An Experience in Relationships as Measuring Rate**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 minutes MATERIALS NEEDED: Large envelope on each table with Modified L1Example 2 inside, enough copies for each participant at the table Stopwatch for facilitator Lesson 1 Paper Passing Recording Chart for Doc Proj Lesson 1: An Experience in Relationships as Measuring Rate Sample Exercise: Paper Passing Directions One participant at each table, take stack of papers out of large envelope labeled A. On my command, take one and pass the remaining stack to the left. Continue passing until all participants have one paper. Last person Stand Up when all participants have a paper (remain standing until further instructions). 15 min Say: Now let’s dive in and experience first hand what students will do in their first day of this Grade 7 course. Read through instructions from slide. Any questions? Complete activity, noting the time for all tables to finish (last person standing at each table). Say: ___ seconds. Not bad. Let’s see if we can get them turned back in to the 1st person in __ seconds. Last person starts, reverse process. Put your paper on the stack and pass right when I say go. New last person stands up when the stack is back together. Get Ready, go. Record time on stopwatch. Use Doc. Camera to record in chart. __ seconds. Excellent, now back in in __ seconds. Get ready, go. Announce time and record in chart. Last time, pass them out. Get Ready, go. How did we measure our rate of passing out papers? Using a stopwatch or similar tool to measure the number of seconds taken to pass out a given number of papers. What quantities will we use to describe our rate? Number of papers passed out and time that it took Describe the quantities you want to measure by talking about what units we use to measure each quantity. One quantity measures the number of papers and the other measures the number of seconds Complete chart with number of papers and time for each trial (if not already completed). Review terms from grade 6 <see next slide> Focus on Ratio for now. Then complete Ratio column of table. When we started passing out papers the ratio was 24 papers in 12 seconds and by the end the ratio was 24 papers in 8 seconds. Are these two ratios equivalent? Why or why not? Guide “students” in a discussion about the fact that the number of papers was constant and the time decreased with each successive trial. The purpose is to see if students can relate this to rate and ultimately determining which rate is greatest? The ratios are not equivalent since we passed the same number of papers in a shorter time. We passed 2 papers per second at the beginning and 3 papers per second by the end. Equivalent ratios must have the same value In another presentation, participants were able to pass 28 papers in 15 seconds, then 28 papers in 12 seconds. A third group passed 18 papers in 10 seconds. How do these compare to our group? (sample data used here) We could find how many papers per second to make these comparisons. Answers on how they compare would vary depending on class results in table. <Review rate terms from next slide.> Then complete rate columns in the table. <Skip to slide 14 for reflection>

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**Ratio and Rate from Grade 6**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: Ratio and Rate from Grade 6 Ratio: A pair of numbers Value of a Ratio: Students described the fraction A / B associated with the ratio A : B as the value of the ratio A to B. Rate: A ratio of two quantities Unit rate: The value of the ratio Rate’s unit: The label, e.g. mph Equivalent Ratios: Two ratios 𝐴:𝐵 and 𝐶:𝐷 are equivalent ratios if there is a positive number, 𝑐, such that 𝐶=𝑐𝐴 and 𝐷=𝑐𝐵. Students understood equivalent ratios to have the same value. 1 min < reference Ratio for paper passing activity, then go back and complete ratio column in chart>

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**Lesson 1: An Experience in Relationships as Measuring Rate**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: Lesson 1: An Experience in Relationships as Measuring Rate Example 1: Paper Passing Why this problem? Turn to your neighbor and answer the following question. Then we will share with the whole group in. Allow 2 min for table discussion then 2 minutes to share. WHY this problem?? Make sure the following is noted: We chose this problem as the first problem kids encounter on day 1 since it is real, it supports creating strong procedures in the classroom, and has a purpose in the math lesson - connecting the math concept to real-world experiences and functions from the get-go. The hope is to get students to recognize that the math we do can have a purpose, that we want to use math to help us with real-world and mathematical problems. Our goal is to get kids thinking within the context of an interesting problem. Instead of telling them we need to find rate to compare, we give them a situation (vary the number of papers or time) and ask them to compare. This leads them to think about how they will do that…guiding them to see why we need the rate, or the unit rate.

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**Ratio and Rate from Grade 6**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Ratio and Rate from Grade 6 Ratio: A pair of numbers Value of a Ratio: Students described the fraction A / B associated with the ratio A : B as the value of the ratio A to B. Rate: A ratio of two quantities Unit rate: The value of the ratio Rate’s unit: The label, e.g. mph Equivalent Ratios: Two ratios 𝐴:𝐵 and 𝐶:𝐷 are equivalent ratios if there is a positive number, 𝑐, such that 𝐶=𝑐𝐴 and 𝐷=𝑐𝐵. Students understood equivalent ratios to have the same value. 3 min Clarify definitions… Say: Many of you may remember how these terms were discussed at May NTI, and how this lays a foundation for this module and may be a different way of thinking about these terms for many teachers. (As a teacher, this was a big change for me to digest. My old thinking… and recognize the new thinking as stated in the progressions and why it is important to make the change for students. …note specificity of language, progressions, foundations for future coursework. As we were writing lessons, we were constantly revisiting the progressions and trying to use mathematically precise language in a student friendly manner. ) Lesson 1 provides an opportunity for students to use the context of the paper passing problem to recall the meanings of these terms, in preparation for the study of proportional relationships. Note: To define the most general form of a rate precisely requires calculus, a concept that is well beyond middle school. While teachers in 6th and 7th grade should describe and speak about the idea of rates and how rates can vary over time, rate should be left undefined for students. Even a precise definition of average rate requires that students know the definition of a function.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Each participant needs the handout from large envelope- Lesson 1, Paper Passing Activity (already passed out in previous activity) Size of whole group and breakdown of teacher vs. non-teacher Document Projector Meet Your Table Talk to your table to determine who is currently teaching in a classroom. Write a ratio of teachers to non-teachers to reflect the participants at your table. Enter this information in the first line of your table. 8 min total Say: Now we are going to use the paper that you passed so many times. Talk to your table to complete the following activity. Complete the first row of the table. For the sake of this discussion, a teacher is currently teaching in the classroom and a non-teacher just means that you are not currently teaching in a classroom – we recognize that some participants may hold multiple roles or certifications (admin, teaching, etc.). Any questions? Give participants 3 minutes to share and complete 1st row of table. Elicit responses, 1 per table, to complete chart as a whole group. Note to facilitator: Devise a method of collecting data to know what number each table is labeled in chart. Include discussion points: Are the ratios of teachers to non-teachers in the first two tables equivalent? What could these ratios tell us? What about the ratio of teachers to non-teachers in table 1 to the entire group. Are they equivalent? If there is a larger ratio of teachers to non-teachers in one table than in the group as a whole, what doe that mean must be true about the ratio in other tables? Other tables would have a smaller ratio. Use document projector to explain this with a table if needed. How do we compare the ratios when we have varying sizes of quantities? Inform participants that when this is done in the class, teachers would need to have this whole group/school information ahead of time. Since students will be comparing from one class to another, we also include a piece that challenges students to use the ratios and the whole grade information to find missing amounts. Say: Finding unit rate may help. In the data given here, the unit rate for table 1 is ___, and the unit rate for whole group is approx.. ___. The unit rate for table 2 is approx._____, etc.. This extension would also allow for students to see the usefulness of using unit rate when making comparisons. We designed lesson 1 so that 7th graders get started thinking about math on day 1 and using the math to think about the world around them.

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**Lesson 1: An Experience in Relationships as Measuring Rate**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 6 minutes MATERIALS NEEDED: Lesson 1 Student Materials (full lesson) Lesson 1: An Experience in Relationships as Measuring Rate Content Scan the lesson to identify what content is addressed in the problems for this lesson. What are kids being asked to do? Turn and talk with others at your table about your observations. 6 min Turn to page __ where you will find Lesson 1 student materials. Scan the lesson to identify what content is addressed in the problems for this lesson. Hopefully you will see some of the things we have already mentioned. After we discuss the content you find here, we will look at the development and function of each lesson component. We will share out your findings in 5 minutes. Example 1(already completed)- writing and comparing ratios, rates, unit rates Example 2- writing equivalent ratios Exercise 1- finding better buy (comparing unit price) Problem Sets- finding rate and unit rate, writing equiv. ratios, identifying if ratios are equiv. , unit price So you can see how it is so important that students have a clear understanding of this language we stopped to talk about previously.

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**Lesson Organization Teacher Student Outcomes**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 6 minutes MATERIALS NEEDED: 1 copy for facilitator of lesson 1 student and teacher Document projector Lesson Organization Teacher Student Outcomes Lesson Notes (in select lessons) Classwork General directions and guidance, including timing guidance Discussion points with expected student responses Student classwork with solutions Scaffolding Boxes Exit Ticket Problem Set (with solutions) 6 min Direct participants to student lesson 1 on pp. x-x and the corresponding teacher lesson on pp. x-x. Demonstrate each feature on the document camera, starting with student materials. Say: We wanted to show you how the lessons are organized before digging more deeply into the mathematics. This slide shows the features in the student and teacher versions of each lesson. Paraphrase or say the following- Opening Exercise: Not all lessons have this feature. If the feature is missing then the first exercise or example builds in the previous day’s work or starts out setting the stage for more in-depth coverage of the lesson content. Classwork: Depending on the lessons, these could be examples, exercises, exploratory challenges or discussions. Also in the classwork pages you will see key definitions if appropriate and on some lessons a lesson summary box. Students are expected to respond to the classwork on their lesson handout. Problem Set: Each lesson includes a problem set. Students are expected to work these problems on a separate sheet of paper. Teacher Materials: In the teacher materials, the lesson type is identified and student outcomes are listed. Then there is commentary and teaching suggestions for the classwork. Scaffolding boxes provide opportunities and suggestions for differentiation of instruction. There are exit tickets for each lesson that assess the student outcomes. Solutions are provided for classwork, exit tickets and assorted problem from the problem set. Pause to take a look at the closing in the teacher materials (on the doc projector). Say: The closing is like the pinnacle of your lesson. If you are short on time – this is what you protect at the expense of other parts. Includes sample dialogue or suggested lists of questions to invite the reflection and active processing of the totality of the lesson experience. Encourages students to articulate the focus of the lesson and the learning that has occurred. Promotes mathematical conversation with and among students. Student Classwork Problem Set

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**Lesson 1: An Experience in Relationships as Measuring Rate**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Participants need Lesson1 exit ticket in handout packet Lesson 1: An Experience in Relationships as Measuring Rate Exit Ticket Tillman the English Bulldog Insert video link 5 min total Say: Now that we have had a chance to see the lesson layout and start thinking a little about ratios and rates, let’s get back to doing some math and actually trying out an exit ticket. Watch the video. After one viewing, Say: What is the length given in the video. 100 m. Students might need to know that is just a little longer than a football field- which is 100 yds. Have participants turn to the Exit Ticket on page ___ of the handout packet and read the questions. Then show video again. Say: Watch it again and then complete the exit ticket in handout your packet. Feel free to take a look at the expected student response in the teacher materials after you are finished. ( 3 min)

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**Exit Ticket Reflection**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Exit Ticket Reflection What do you notice about the exit ticket? How does it compare to what you have used in the classroom or have seen teachers use in the past? Take 3 minutes to talk with your table. 5 min Read question from slide. Allow 3 min for participants to share at table then ask for responses. Ensure the following points are noted. Interesting for students Direct connection to student outcome(s) and standards. Not only are we asking for students to find a unit rate but they need to know to use it when comparing multiple quantities that may be given in varying amounts. How do you know? At the end of questions students are asked to justify their thinking. Teachers can use the exit ticket as formative assessment to adjust plans accordingly. It could be a refresher the next day if not enough time, or to repeat exact question again to see retention. Self-check: Good check in for students “did I learn what was intended in this lesson? “ before they get home and stumble on HW

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: Lesson 2: Proportional Relationships This lesson introduces the notion of Proportional Relationship Serves as the connecting piece from Grade 6 to Grade 8 Grade 6 Ratios Equivalent Ratios Rate and Unit Rate Grade 7 Proportional Relationship Unit Rate as Constant of Proportionality Grade 8 and up Linear Relationships Study of Functions In grade 6, students wrote ratios and were able to determine whether ratios were equivalent. In grade 7, we can now talk about when two sets of quantities are related (paired) with each other to define a proportional relationship. Initially in the lessons, the term “proportional relationship” is suppressed and instead a much simpler term, “proportional to” is used. The ratio A:B, or the opposite associated ratio B:A can be used to determine a proportional relationship.

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Lesson 2: Proportional Relationships Classwork Example 1: Pay by the Ounce Frozen Yogurt! A new self-serve frozen yogurt store opened this summer that sells its yogurt at a price based upon the total weight of the yogurt and its toppings in a dish. Each member of Isabelle’s family weighed their dish and this is what they found. *Fix formatting of table* Turn to page __ of your handout to find Lesson 2 Example 1. Say: We are going to go through a couple of examples modeling the teacher portion if you will take a moment to step back into a student’s shoes. Take 1 minute to read the problem. Discussion: Does everyone pay the same cost per ounce? Yes, it costs $0.40 per ounce Isabelle’s brother takes an extra-long time to create his dish. When he puts it on the scale, it weighs 15 ounces. If everyone pays the same rate in this store, how much will his dish cost? How did you calculate this cost? $6. Guide “students” to notice that if you multiply the number of ounces by the constant (cost per ounce), it will give you the total cost. Take a moment to have students confer that this would be true for the values they found. Since this is true, we say” cost is proportional to weight”. You may have noticed there is a place in the student materials for students to complete this statement. <Click for animation> What happens if you don’t serve yourself any yogurt or toppings, how much do you pay? $0. And does the relationship above still hold true? In other words, if you buy 0 ounces of yogurt, then multiply by the cost per ounce, do you get 0? So, even for 0, you can still multiply by this constant value to get the cost (not that you would do this but we can examine this situation for the sake of developing a rule that is always true). <click to advance and show table with the relationship modeled in table> Weight (ounces) 12.5 10 5 8 Cost ($) 4 2 3.20 Cost ____ ______________________ ____ Weight. is proportional to

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: Lesson 2: Proportional Relationships Classwork Example 1: Pay by the Ounce Frozen Yogurt! Continue Discussion: So, you always multiply the number of ounces, x , by the constant that represents cost per ounce, to get the total cost, y. Pause with students to note that any variables could be chosen but for the sake of this discussion, they are x and y. For any measure x, how do we find y? Multiply it by 0.4(unit price). Indicate this on the given chart as done below. Be sure students do the same. Say, Big Progression here… Students are able to identify a pattern (that all costs are proportional to their corresponding number of ounces), then use that pattern to find a new cost, test the pattern with an unusual case (0 ounces for 0 dollars), and then generate a rule from these steps. Finally, they generalize the rule and walk away having built up to the key understanding of the lesson. • 0.4

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lesson 2, Example 2 handout. Lesson 2: Proportional Relationships Example 2: A Cooking Cheat Sheet! In the back of a recipe book, a diagram provides easy conversions to use while cooking. Ounces ____ ______________________ ____ Cups. Exercise 1 During Jose’s physical education class today, students visited activity stations. Next to each station was a chart depicting how many Calories (on average) would be burned by completing the activity. Calories burned while Jumping Rope Time (minutes) Calories Burned Is the number of Calories burned proportional to time? How do you know? 2 min Following from that first example, let’s take a quick look at this same idea represented with a double number line. Turn to the next Example, Lesson 2, Example 2 in your handout. What does the diagram tell us? The number of ounces in a given number of cups. More specifically, each pair of numbers indicates the correct matching of ounces to cups. Is the number of ounces proportional to the number of cups? How do you know? There are 8 ounces for every cup and to get the number of ounces, you can always multiply the number of cups by 8. Have students complete statement on page, ounces is proportional to cups and note how they can tell. It is important to acknowledge that you could also divide by 8 if you know the number ounces and are trying to find the number of cups. This discussion should lead to the importance of defining the quantities clearly. How many ounces are there in 4 cups? 5 cups? 8 cups? 32, 40, 64 For the sake of this discussion (and to provide continuity from between examples), let’s represent the cups with x, and the ounces with y. Teacher should label the diagram with the indicated variables and guide student to do the same. For any number of cups x, how do we find the number of ounces, y? Multiply it by 8. So y = 8x.

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Lesson 2: Proportional Relationships Exercise 1 During Jose’s physical education class today, students visited activity stations. Next to each station was a chart depicting how many Calories (on average) would be burned by completing the activity. a. Is the number of Calories burned proportional to time? How do you know? b. If Jose jumped rope for 6.5 minutes, how many calories would he expect to burn? 5 min total Turn to pg. __ in our handouts for Lesson 2, Exercise 1. Now your turn to try one on your own. Spend 2 minutes completing Exercise 1 What kind of responses do we want students to come up with here? Share answers with your elbow partner. (2 min). Give me a thumbs up if you said any of the following: We want them to be able to apply what they learned so far in the lesson. (formative assessment) We want them to be able to explain how they arrived at their answer. We want them to apply to an extension of the same situation. We have designed lessons intentionally so that it follows directly from Example 1 and Example 2. Students will apply the discovery from Example 1 together with the representation (double number line) from Example 2 as they work independently on Exercise 1. Note, answers to problems here if needed: Is the number of Calories burned proportional to time? How do you know? Yes, the time is always multiplied by the same number, 11, to find the calories burned. If Jose jumped rope for 6.5 minutes, how many calories would he expect to burn? 71.5 since 6.5 times 11 is 71.5.

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**Lesson 2: Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lesson 2: Proportional Relationships Closing How do we know if two quantities are proportional to each other? Two quantities are proportional to each other if there is one constant number that is multiplied by each measure in the first quantity to give the corresponding measure in the second quantity. How can we recognize a proportional relationship when looking at a table or a set of ratios? 2 min Say: What do we expect students to walk away with? What are the expected responses here? Take 1 min to share at your table. Here are the responses that indicate students have reached the desired outcomes. < click to advance samples> Say: We are focusing on the term proportional to as it is easier for students to digest than proportional relationship. It also fits well in problems and everyday language, the number of dollars earned is proportional to the number of hours worked, or distance is proportional to time. Once students have an understanding of the concept, then we introduce the proportional relationship language, but continue not to make a big deal about it. 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. If each of the measures in the second quantity is divided by its corresponding measure in the first quantity and it produces the same number, called a constant, then the two quantities are proportional to each other.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Lesson 3 Opening Exercise in Handout packet for participants Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in Tables Lesson 3 Opening Exercise: You have been hired by your neighbor to babysit their children Friday night. You are paid $8 per hour. Complete the table relating your pay to the number of hours you worked. Based on the table above, is pay proportional to hours worked? How do you know? 5 min Turn to the Opening Exercise for Lesson 3 on pg. __ of your handout packet. Take 1 min to read. What are your initial observations about this opening exercise? Ask for participants to share. (1 min) Take 2 minutes to complete the problem. How do you think students will respond to this exercise? (1 min)

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lesson 3 Examples 1 and 2 Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in Tables Lesson 3: Compare and contrast Examples 1 and 2. Share your observations with a neighbor. 2 min Turn to page __ in Student materials. (Lesson 3 examples 1 and 2) Take 1 minute to talk with a neighbor. Ask for participants to share what they noticed with the whole group. Similarities: Both present data in a table and ask students to decide whether it represents a prop. relationship. Both are proportional. Both give a context. Both use x and y (We use mostly x and y at first until students solidify the conceptual understanding then we start to vary the letters that we choose for variables. ) Difference: In Example 1, the x-value is smaller than the y value, whereas Example 2 has an x-value larger than the y value. (This allows for students to see where the constant of proportionality is greater than 1 and less than 1. )

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: x minutes MATERIALS NEEDED: Lessons 3-4: Identifying Proportional and Non-Proportional Relationships in Tables Lesson 4 Example: Which team will win Randy’s race? Take 10 minutes to complete all parts of the problem independently. *skip if time is an issue* Turn to page __ (lesson 4 example) Take 10 minutes to complete this example on your own, then I will ask you to collaborate with those at your table. Click for animation- 1st for table share, for table share graphic Ask for participants to share approaches as problem is discussed as whole group. Use document projector to show some of the approaches that are shared. Encourage multiple approaches. Take 5 minutes to share your solutions at your table. Be sure to explain your approach to the problem. Be prepared to share your approach with the whole group.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED: Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs Classwork Opening Exercise: Isaiah is selling candy bars to help raise money for his scouting troop. The table shows the amount of candy he sold to the money he received. Is the amount of chocolate bars sold proportional to the money Isaiah received? How do you know? x Chocolate Bars Sold y Money Received ($) 2 3 4 5 8 9 12 Turn to page __ to find lesson 5 student materials. Take 3 minutes to answer the question independently. At the end of the time, ask for participants to share responses: The two quantities are not proportional to each other because a constant describing the proportion does not exist. Hopefully you found that this exercise builds upon what students have learned in previous lessons and guides them into developing the characteristics of a proportional relationship on a graph. Now let’s take a look at how they will do this.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Graph paper with table drawn and axes set up under doc projector Ruler for participants and facilitator at doc projector Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs x y 2 3 Example 1: From a table to graph Create another ratio table that contain two sets of quantities that are proportional to each other using the first set of ratios on the table. Give participants 2 minutes to create a table of ratios, then ask for responses. Discussion: Do we all need the same points in our table? How do we get these points on a graph? Students may need a quick recap (from grades 5 and 6) on coordinate plane, x- axis, y-axis, origin, quadrants, plotting points, and ordered pairs. Why did I only set up quadrant I? the measures in our ratios (number of candy bars, amount of money made) will be positive (we hope ) Graph some of the points participants share, noting that with students we would want to label axes, do we use x or y for each quantity? Why? x axis should be the chocolate bars and the y axis should be the amount of money received. The amount of money received should depend on the candy bars being sold so the amount of money should be y, the dependent variable. What observations can you make about the arrangement of points? The points all fall on a line. Use a ruler to join the points. Does it make sense that we are connecting the points? Would we sell part of a chocolate bar? Do we extend the line in both directions? Why or why not? Technically the line for this situation should start at (0,0) to represent 0 dollars for 0 chocolate bars and extend infinitely in the positive direction because the more chocolate bars Isaiah sells, the more he makes. Would all proportional relationships pass through the origin? Share some of the context of previous examples and whether (0,0) would always be included on the line that passes through the pairs of points in a proportional relationship. Yes, it should always be included for proportional relationships. For example, thinking back to the first example of buying frozen yogurt by the ounce, if a customer buys 0 ounces of yogurt, then he/she would pay zero dollars. We are guiding students to develop the characteristics of the graph and then complete the table with what they have discovered. The points will fall on a straight line that goes through the origin. Notice the specificity of language here…the graph is not a line (due to the fact that not all the points between those we plotted would be make sense in this situation- therefore not part of the proportional relationship) but is shows that all the points would fall on a line that passes through the origin. It might even be that (0,0) would not make sense to be part of some proportional relationships, depending on the context, but the line would always pass through this point. Good opportunity to talk with kids about why it is a straight line. Each chocolate bar is being sold for $1.50 each, which is the unit rate and also the constant of the proportion. This means that for every increase of 1 on the x-axis, there will be an increase of the same amount (the constant) on the y-axis. This creates a straight line. **Highlight how MP.1 is addressed in this example. You have probably done or seen similar things done in classrooms. From my experience, kids always seemed to struggle with the whole concept of direct variation. In the study of linear relationships, we would just throw in a day or two on direct variation and the kids would always say, a what? I found that the graph always helped them to make sense of this type of prop relationship (of course I never really called it that). Now we have already spend 4 lessons previously looking at this type of relationship and now they are seeing the graphical representation and connecting it back to contextual situations to make sense of why it looks the way it does. Providing that context gives kids something to connect to, to pull all the representations together. As a teacher, I appreciate how the standards are finally allowing us to delve into the concepts and make sure kids really know what they are talking about. Important Note: Characteristics of graphs of proportional relationships:

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs What is a common mistake a student might make when deciding whether a graph of two quantities shows that they are proportional to each other? How might we counteract this common misconception? Read question to participants. Allow for participants to share at their table for 1 minute, then ask for responses. They might see that they fall on a line and assume that the quantities are proportional to each other. Our goal with this curriculum: In the following examples and exercises as well as the problem set, we will have students exploring different graphs, some of which are lines but don’t pass through the origin, and some of which are nonlinear but do pass through the origin to try to have them recognize the different situations that may exist to solidify the specificity of the language that is necessary in describing the graph of a proportional relationship. (When I taught this just this past year, I found that kids did a pretty good job at memorizing this description of the graph, however I was never satisfied with that alone. Our hope is that by going through this discovery process on their own, along with the lessons leading into this, students will understand why it appears this way on the graph, and be able to connect it to the verbal description or context and the table or diagram. )

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 minutes MATERIALS NEEDED: Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs Table Activity: Look in Envelope 6 and take 10 minutes to discuss the given problem and have one table member record responses on the chart paper. At the end of the time, we will post the chart paper on the wall and have an opportunity to participate in gallery walk. Lesson 6 is an Exploration Type Lesson where students are working in groups to use multiple methods to show whether the quantities represented are proportional to each other or not. Participants have 10 minutes to discuss the given problem ( “monster sundae” and “colonies of mold”) and record their responses onto the poster paper. For the last 5 minutes, have groups adhere their posters on the wall and circulate around the room writing thoughts on sticky notes to adhere to posters. Have participants with the same ratios identify and discuss the differences of their posters. At the end of the gallery walk time, whole group will come together and presenter will circulate among posters to pull some of the sticky note comments to discuss from posters.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Lessons 5-6: Identifying Proportional and Non-Proportional Relationships in Graphs Gallery Walk Use Sticky Notes to… Note differences found in groups who had the same ratios. Note any common mistakes and how it could be fixed. Note any a poster that stood out that represented their problems and findings exceptionally clear. *edit slide* At the end of the gallery walk time, whole group will come together and presenter will circulate among posters to pull some of the sticky note comments to discuss from posters. Say: The problems included in this portion of Lesson 6 show a variety of relationships, including some potentially confusing ones such as the linear graph that does not pass through the origin and the quadratic function. These have been intentionally included for students to grapple with because there will be evidence to present on both sides of the argument and will hopefully provide for some robust debate within the classroom. Take a look at the closing to the day 6 lesson, find the closing questions in the teaching materials on page ___. Closing Questions with students would pull together observations from the posters and identify common components of accurately described explanations and characteristics of visually appealing and informative graphs. Exit ticket serves as a final reflection on the characteristics of proportional graphs.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Conclusion of Topic A Take a few moments to reflect upon how you will be able to promote successful implementation of these lessons in your classroom, school, district, and/or BOCES? What do you think teachers would need to know? Take 3 minutes to reflect individually, then share with your table. (8 minutes total)

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Topic Opener In Topic B, students learn to identify the constant of proportionality by finding the unit rate in the collection of equivalent ratios. They represent this relationship with equations of the form y = kx, where k is the constant of proportionality (7.RP.2, 7.RP.2c). In Lessons 8 and 9, students derive the constant of proportionality from the description of a real world context and relate the equation representing the relationship to a corresponding ratio table and/or graphical representation (7.RP.2b, 7.EE.4). Topic B concludes with students consolidating their graphical understandings of proportional relationships as they interpret the meanings of the points (0,0) and (1, r), where r is the unit rate, in terms of the situation or context of a given problem (7.RP.2d). Turn to __ to find the topic opener for Topic B. What will students be learning in this topic? Note the focus standards, the number of instructional days (_) , and description of lesson concepts

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**Lesson 7: Unit Rate as the Constant of Proportionality**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lesson 7: Unit Rate as the Constant of Proportionality Haven’t we already learned about the Constant of Proportionality? Why is this lesson here? Yes! Our module was designed so that students would be able to describe a proportional relationship by looking at the relationship from one quantity to another. They started back in Lesson 2 identifying whether there exists a number (the same number) that relates x to y or y to x. At the time, we focused on finding out if a number existed. We felt it was important for students to develop the conceptual understanding before overwhelming them with this vocabulary. Now, for the past 5 lessons, we have already been talking about this “same value” that exists in a proportional relationship. We looked at it from a table and on a graph. Now we are finally giving it a name and relating it back to the context. Find the constant of proportionality and describe what it means in the context of the situation. So our focus here is on connecting it back to the context and getting students to realize that it is just the same thing as unit rate. See our closing questions on the next slide…

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**Lesson 7: Unit Rate as the Constant of Proportionality**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: Lesson 7: Unit Rate as the Constant of Proportionality Closing Questions What is another name for the number that relates the measures of two quantities? (How do I get from x to y?) How is the Constant of Proportionality related to the unit rate? Constant of Proportionality It is the same. Both tell us how many units of one quantity for every 1 unit of a second quantity. Once we see it used here as the constant of proportionality, it might make more sense why unit rate is the numerical value of the rate. (As a writer (and teacher), it wasn’t until now that I finally started to get that piece. I had read and re-read the progressions regarding unit rate and accepted it for what it was, but then when we started to connect it to the constant of prop that is used in the equation, it all came together for me. And of course, that is what we are hoping to relay to our students!)

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**Lesson 7: Unit Rate as the Constant of Proportionality**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Lesson 7: Unit Rate as the Constant of Proportionality Exit Ticket Susan and John are buying cold drinks for a neighborhood picnic, where each person is expected to drink 1 can of soda. Susan says that if you multiply the unit price for a can of soda by the number of people attending the picnic, you will be able to determine the total cost of the soda. John says that if you divide the cost of a 12-pack of soda by the number of sodas, you will be able to determine the total cost of the sodas. Who is right and why? *Could skip if short on time* Take 3 minutes to answer the ET problem here from Lesson 7.

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**Lesson 7: Unit Rate as the Constant of Proportionality**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minutes MATERIALS NEEDED: Lesson 7: Unit Rate as the Constant of Proportionality Exit Ticket Solution *insert typed answer in slide, once revised*

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**Lesson 8-9: Representing Proportional Relationships with Equations**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 minutes MATERIALS NEEDED: Lesson 8-9: Representing Proportional Relationships with Equations Take 15 minutes to go through Lesson 8, completing Examples 1 and 2 and any 2 questions from the Problem Set. Feel free to compare your answers with what is provided in the teacher materials. Turn to page _ in the Student Materials- Lesson 8. Normally, examples are teacher-led but for the sake of our presentation, we are going to have you try them independently and be prepared to talk about this lesson and how the components are connected.

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**Lesson 8-9: Representing Proportional Relationships with Equations**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 4 minutes MATERIALS NEEDED:- Lesson 8-9: Representing Proportional Relationships with Equations Discuss at your table: What is the main idea of this lesson? How do the Examples prepare students to find success with the Problem Set? Give tables 3 minutes to discuss then invite responses.

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**Lesson 8-9: Representing Proportional Relationships with Equations**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: Lesson 8-9: Representing Proportional Relationships with Equations Lesson 9 Exit Ticket Oscar and Maria each wrote an equation that they felt represented the proportional relationship between distance in km and distance in miles. One entry in the table paired 150 km with 93 miles. If k = number of kilometers and m= number of miles, who wrote the correct equation that would relate miles to kilometers and why? The purpose of this ET is to see if kids can work past a misconception that often occurs with this concept. It is assessing whether students can apply what was discussed in lesson to decipher the intended response. For the purpose of this activity, I removed the equation options to see if you could figure out where the misconception might lie. Take 5 minutes to see if you can come up with how Oscar and Maria both wrote the equation differently. You will then have 1 minute to discuss with an elbow partner. I will invite you to the document projector to share your responses after you discuss. The two options (names don’t matter)… Oscar wrote the equation k=1.61m and he said that the rate 1.61/1 represents miles per km. Maria wrote the equation k=0.62m as her equation and she said that 0.62 represents miles per km. The second one above is correct. Maria found the unit rate to be 0.62 by dividing miles by km. The rate that Michael used represents km per mile. However, it should be noted that the variables were not well-defined. Since we do not know which values are independent or dependent, each equation should include a definition of each variable. For example, Maria should have stated that k represents number of km and m represents number of miles.

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**Lesson 10: Interpreting Graphs of Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 21 minutes MATERIALS NEEDED: Lesson 10: Interpreting Graphs of Proportional Relationships Spend 3 minutes previewing Lesson 10 materials. What are your initial observations about this lesson? Spend 3 minutes previewing Lesson 10 materials. Make sure the following points are noted: Lesson 10 pulls together student understanding of the graphical representation of a proportional relationship. Students interpret the meanings of the point (0,0) and (1,r) in terms of the context of a situation. Click animation. Allow participants 15 minutes to complete Examples 1 and 2. Talk with a neighbor about your responses and how you think students will respond to this lesson. I’ll ask for your thoughts in 3 minutes. Take 15 minutes to complete Examples 1 and 2. How do you anticipate students will respond to this lesson?

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: X Biggest Takeaway What is your biggest takeaway from this session? Think for just 1 minute. “Whip Around” the table to share. 5 minutes total Say: Take one minute to reflect on this session. What, for you, is the biggest takeaway? Jot down your thoughts. Then you will have time to share. Give participants 1 minute for silent, independent reflection. <CLICK TO ADVANCE ANIMATION ON SLIDE> “Whip Around” at your table about your reflections. 2 minutes Then, facilitate a discussion that leads into the key points on the next slide. 2 minutes Say: Please write ALL questions, concerns, comments on post its and leave on parking lot. We will review them all this evening and be better prepared to support you tomorrow.

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: X Key Points Lessons are designed for students to be active learners. Teacher questioning guides students to explore ideas, investigate natural questions, and develop the big ideas that are the focus of the module. They are uncovering the story. Students use the study of ratios in Grade 6 as a foundation to examine the characteristics of a proportional relationship in Grade 7. Students make connections between multiple representations of proportional relationships (verbal, diagram, table, graph) using each one to develop a deeper understanding of what really make a proportional relationship. 2 min Read through the bullet points to emphasize the key points from the session. 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.

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**A Story of Ratios G7-M1 : Mid-Module Assessment**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: A Story of Ratios G7-M1 : Mid-Module Assessment This afternoon we are going to focus our attention on the Mid-Module Assessment for Grade 7 Module 1. Is there anyone present this afternoon who was not here for the morning session? (Call for a show of hands.) So what do we hope you are able to do as a result of this session?

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Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 30 seconds MATERIALS NEEDED: Objectives To articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment Our universal goal is for you to be able to determine some key instructional practices to implement in your classroom, school, school district, and/or BOCES that will prepare students to sufficiently take on these types of assessment questions. Let’s take a look at our agenda for this afternoon

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**Agenda View and complete the Mid-Module Assessment.**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 1 minute MATERIALS NEEDED: Agenda View and complete the Mid-Module Assessment. Discuss the rubric and use it to score each other’s assessment. Share notable examples of work and scoring. Round-table and whole group discussions at various times in the session related to the assessment questions and scoring. I know that as educators we have all sorts of questions in the back of our minds that go beyond the assessment. I ask that today you try to view and complete this Mid-Module assessment from a student’s perspective. Please note that we will provide you with a separate copy of the assessment so that the copy in your binder can remain clean for future photocopying and turn-key training. We will discuss the scoring rubric so that you understand its make-up, and then we will use it to score each other’s assessment. You will be given opportunity to share out and discuss your notable examples of work and scoring. And throughout the session we will have discussions at your table and whole group with regard to the assessment questions and scoring. 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?

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**G7-M1: Ratios and Proportional Relationships**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 seconds MATERIALS NEEDED: G7-M1: Ratios and Proportional Relationships Our focus is on the Mid-Module assessment which is the first assessment in grade 7 Module 1: Ratios and Proportional Relationships.

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**G7-M1 Mid-Module Assessment**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 seconds MATERIALS NEEDED: G7-M1 Mid-Module Assessment Please turn to the table of contents of the Module Overview in your binder. You will notice that the Mid-Module Assessment follows lessons 1-10, and those topics comprise Topics A and B.

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**Standards Assessed Which Topics and Standards will be assessed?**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 3 minutes MATERIALS NEEDED: Standards Assessed Which Topics and Standards will be assessed? In the Module Overview, look through the lessons and standards that are referenced in lessons Then look at the Assessment Summary at the end of the Module overview. Take a couple of minutes (2-minutes) and discuss with your neighbors the topics (and standards) that you would expect to see on this Mid-Module Assessment. (After 2 minutes) Ask participants to share out answers with the whole group. The participants should identify: Topic A: Proportional Relationships (7.RP.2a) Topic B: Unit Rate and the Constant of Proportionality (7.RP.2b, 7.RP.2c, 7.RP.2d, 7.EE.4) (Presenters begin distributing copies of the assessment to participants, so that they can keep their binder copy clean for future turn-key training and photocopying).

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**Understanding the Assessment**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 16 minutes MATERIALS NEEDED: G7-M1 Mid-Module Assessment Copies Pencils Understanding the Assessment I’d like you to spend the next 15-minutes completing the Mid-Module Assessment. Please remember to complete the assessment from the perspective of a student. This will help you and the scorer better understand the rubric and scoring process.

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**A Progression Toward Mastery**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 minutes MATERIALS NEEDED: A Progression Toward Mastery Please turn to the rubric and student work for the assessment. The assessment rubric is designed to help categorize students responses into 1 of 4 categories of Progression Toward Mastery. Please note that steps 1-4 are not assigned point values that you will necessarily add up when scoring the assessment; rather they are for assessing the student’s level of mastery of concepts. To help make sense of this, I will clarify what each step indicates. Step 1 {I’m not getting it}: Represents a bare minimum of knowledge but there is some presence of valid reasoning. Step 2 {I’m starting to get it}: Represents a mid-level, developing knowledge of the concept but does shows some room for growth Step 3 {I’ve got it!}: Represents an upper-level knowledge in which a student shows that they know what they’re doing. Step 4 {I can teach this!}: Represents mastery in which a student not only knows what they are doing, but can provide reasoning and valid justification for their solution.

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**Scoring Student Work TIME ALLOTTED FOR THIS SLIDE: 10 min 30 sec**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 10 min 30 sec MATERIALS NEEDED: Pens Scoring Student Work Exchange your completed assessment with a neighbor and score their work according to the Progression Toward Mastery rubric and the provided student work. When you are finished, share in conversations at your table notable examples of work and scoring of the assessment according to the rubric. Principals- 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.

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**Sharing Student Work TIME ALLOTTED FOR THIS SLIDE: 10 minutes**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 10 minutes MATERIALS NEEDED: Document Camera Sharing Student Work Have several participant volunteers bring their work up to the document camera and share the score they assigned to it based on the rubric. Time is allotted for questions from the audience and whole-group consensus on scoring specific work.

57
**Standards for Mathematical Practice**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 8 minutes MATERIALS NEEDED: Copies of the 8 Mathematical Practices Pencils Highlighters Standards for Mathematical Practice Locate the Standards for Mathematical Practice listing in your materials. Take a few minutes (7 minutes) to identify the standards for mathematical practice that are addressed in each assessment question. Discuss it as a table, and record the Mathematical Practice number in the margin aside each assessment question.

58
**Evidence of the Mathematical Practices**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 15 minutes MATERIALS NEEDED: Post-it Notes Pens and Markers Posters of Assessment Questions stationed around the room Document Camera Evidence of the Mathematical Practices How and where do they appear in the assessment items? Over the next several minutes I’d like each table to identify the mathematical practice standards that you found in each assessment question and write its number on a post-it note. Then I want you to stick the post-it note(s) on the poster that corresponds with each assessment question around the room. Presenter walks around the room and goes from assessment question to assessment question, stating the Mathematical Practices that were referenced the most according to the post-it notes. At each stop (assessment question poster), the presenter elicits an explanation from a participant. Participants verbally explain why they selected that Mathematical Practice and if necessary, the document camera is used to show where and how they saw the mathematical practices reflected in the assessment item.

59
**Biggest Takeaway TIME ALLOTTED FOR THIS SLIDE: 5 minutes**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 5 minutes MATERIALS NEEDED: Biggest Takeaway Take a minute to brainstorm at your tables and determine your biggest takeaway from today’s session. In a minute or so we’ll ask for a spokesperson at each table to state their group’s biggest takeaway.

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**Key Points Assessment questions TIME ALLOTTED FOR THIS SLIDE:**

Header July 2013 Network Team Institute TIME ALLOTTED FOR THIS SLIDE: 2 min 30 seconds MATERIALS NEEDED: Key Points Assessment questions Are directly aligned to the CCSSM May address multiple learning standards Are formatted to invoke higher level thinking and address MPs Scoring Rubric Progression Toward Mastery Here are the key points that we want you to take away from today’s session. The development of assessment questions is one of the first tasks in developing the modules so the questions are written in direct alignment to the Common Core learning standards. The module lessons are written with the assessments in mind, not the other way around. Assessment questions are formatted such that they access higher order thinking and address the mathematical practice standards. You also noticed that parts of questions addressed multiple standards simultaneously. The assessment scoring rubric is designed to assess a student’s progress toward mastery of concepts and focuses on progress rather than deficiencies.

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