Dear Teacher,



Dear Teacher,As an introduction to this Algebra curriculum we would like you to read Becoming a Wonderful Mathematics Teacher. In this document we attempt to give language to the work a math teacher would do with his/her students in order to engage an move them to a new place in their thinking about mathematics.Throughout this curriculum we will use the language of the 8 Aspects of a Wonderful Mathematics Teacher to help guide you in both your planning and facilitation of the lessons.We would be very interested in your thoughts about these eight aspects and how we can have a yearlong dialogue about their meaning and expression in the classroom.Sincerely,ISA Math CoachesBecoming a Wonderful Mathematics Teachertc "Becoming a Wonderful Mathematics Teacher" \l 2 “… a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting their problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking.” – George Polya (1945)IntroductionMathematics is a subject most people studied in school for 11 or 12 years but many people never came to appreciate or understand it. Why is that? Is it the discipline itself or the way it has been presented to people? Can mathematics be a subject that most students can begin to make sense of and enjoy if they are given meaningful opportunities? The Common Core is making new demands on the teachers of mathematics throughout the country. The demands impact both the content teachers are being asked to teach and the standards of practice they employ in the classroom. An inquiry-based approach offers teachers a means of giving students an opportunity to engage with the new content while also developing as mathematical thinkers. What is inquiry-based instruction in mathematics?Inquiry-based instruction has a long history in both education and mathematics education. In inquiry-based instruction, mathematics is viewed as a humanistic discipline where students construct meaning and understanding within a community of learners (Borasi, 1992). It is a multifaceted approach to learning. Students are encouraged to wonder about mathematical ideas, raise questions, make observations, gather data, consider possible relationships and patterns within the data, make conjectures, test one’s conjectures, and finally generalize a discovery supported by evidence (Borasi, 1992; NRC, 1996; Suchman, 1968; Wells, 1999). Generating an idea or concept and arguing for its authenticity is an essential aspect of inquiry and tells a teacher what a student knows about mathematics (Koehler & Grouws, 1992; Lampert, 1990).The Common Core demands that teachers move away from math being viewed as a set of procedures students learn to replicate and show on an exam to the presentation of mathematics as a coherent discipline that one begins to understand over time so it can be used within multiple problematic situations.Using an inquiry/problem-based approach can help students develop both conceptual and procedural understanding while learning to reason quantitatively and abstractly. Using an inquiry/problem-based approach can help students move from a concrete way of thinking to an abstract way of thinking. This can take the form of moving from an arithmetic way of thinking to an algebraic way of thinking.Using an inquiry/problem based approach helps students develop independence and perseverance. Students learn to take risks, rethink strategies and make sense of process and solution.8 Aspects of a Wonderful Mathematics TeacherIn order to develop students who make sense of the discipline of mathematics they must experience high school mathematics courses that enable them to develop as mathematical thinkers. Teachers of mathematics have before them an opportunity to enter into the wonder-filled world of mathematics with their students in such a way that their students leave thinking differently. Our work focuses on helping you develop in the 8 aspects of what makes for a wonderful math teacher: Learners, Artists, Decision Makers, Questioners, Modelers of Mathematical Thinking, Provocateurs, Coaches, and Reflectors.Learner –Teaching is always a learning experience. The learning takes different forms. A mathematics teacher is always learning about his/her discipline and about different ways of presenting this wonderful discipline to his/her students. Often what teachers are struggling with in the mathematics classroom goes beyond mathematics and into what the students bring with them to the classroom. From life and past math experiences students come to classrooms as complex human beings that cannot be summarized by one dimensional measures or encapsulated by a set of behaviors. This implies that students’ understandings and misunderstandings of mathematics have more nuance than is present on the surface and requires deeper inspection. Thus, teachers need to see themselves as always learning about how their students think and make sense of mathematics.Artist – Teaching is an art. It takes great imagination and creativity. Mathematics teachers need to find their connection and passion for their discipline and find out how to express it within their teaching. Teachers need to ask, “Why do I love mathematics and how do I bring some of that emotion to my students?” Becoming an artist in the classroom takes time. In the process of doing this teachers move along a continuum of creation. Often, teachers start out through imitation. They use the material of others as they try to get their footing. As teachers develop as artist they move between modifying/tweaking material and creating their own. As they develop their craft they begin to create interesting experiences for students that will lead to conceptual and/or procedural understanding. The teacher as artist is always looking for opportunities to engage students with meaningful problems that bring out the students’ creativity and imagination.Decision Maker – Mathematics teachers are always making decisions. These decisions are about curriculum, selecting problems to use with their students, and possible next steps or activities for supporting individual students. Decisions also happen within lessons. Teachers are always making decisions about stepping back or entering in as students grapple with a problem. If they join a student in his/her struggle teachers need to decide if asking a question would be best or if the student needs some explanation. The big question is always, “What will be the best thing I do for this (these) student(s) in this moment so that their mathematical thinking is being enhanced and not stifled?”Questioner – Questions are central to every aspect of a math teacher’s work. When creating units and their accompanying lessons teachers should always ask, “What are the questions that guide this unit and accompanying lessons? Why would these questions help students to deepen their mathematical understanding? Have I created situations within the unit where my students can ask questions?” Questions are also central to the classroom facilitation. The right questions can motivate students to engage in an activity or further engage in solving a complex problem. Asking the right question is part of the art of teaching and teachers need to see themselves as questioners of their own questions.Modeler of Mathematical Thinking – Modeling has always been seen as central to a teacher’s job in the classroom. A problem with the modeling approach that has been used in the past is that the teacher modeling focused on handing students a recipe to follow and then it was the students’ job to imitate. Making the focus of modeling procedures or recipes did not enhance mathematical thinking, but rather stifled it. Modeling needs to have as its purpose the strengthening of student mathematical thinking. Teachers can do this by modeling their own thinking process. They share with the students the questions they ask themselves while they are engaged in the problem solving process. A discussion about the question and process should follow where students then can decide how they want to integrate the teacher’s way of thinking into their own way of thinking.Provocateur – Students need to be put into situations that take them out of their comfort zone and challenges them to expand their thinking and take risks. These types of experiences help students develop a comfort with going to new places in their thinking, and develop perseverance in solving problems. By provoking student thinking, with problems that can be thought about in multiple ways and having multiple solutions, a teacher tries to get his/her students to go deeper into mathematics with more nuance and breadth. Coach – Students come into the high school math classroom with many experiences (often bad) that affect how they feel about mathematics. Thus teachers have to see themselves as working to change students’ productive disposition in the math classroom. Mathematics teachers are both motivators and critics (raise important questions to students when they go off a path.) They have to see themselves as deeply engaged with each student and figuring out ways to support him/her.Reflector – Successful mathematics teachers are reflective teachers. They reflect on both their teaching and their students’ thinking and learning. Reflection takes place while the lesson is going on and after class as means of informing instruction. Reflection is essential for growth and so all math teachers and teams of math teachers need to create a set of structures where teachers have the opportunity to reflect and get feedback from their colleagues.Unit 3: LinearityAligned to the Common Core Standardstc "Unit 3: Linearity" \l 1Unit 3: LinearityDeveloping Mathematical Thinkingtc "Unit 3 Outline" \l 2Essential Questions: How does linearity help me to better understand both mathematics as a discipline and the meaning of mathematics in the world?How do I think about different mathematics problems and develop strategies that are workable?Interim Assessments/Performance TasksLesson 2: Rate of Change of 4Lesson 3: MatchmakerLesson 4: Changing the TableLesson 6: Johnson Family LogoLesson 9: The Movie RentalLessons 11 & 12: Story GraphsLessons 16 and 17: Five Linear problemsLesson 19: The RaceLesson 22: Three Inequality ProblemsFinal Assessment:Final Project/ReflectionWhat will students understand and be able to do at the end of the unit?Students will understand the power and meaning of the different representations of linearity and they will also understand their relationships.Students will be able to move comfortably between table graph and equation in relation to linearity.Students will have greater ability to solve different types of linear problems including systems of equation problems and system of inequality problems.Students will have a more nuanced understanding of the meaning of “solution.’.What enduring understanding will students have?Students will understand the uniqueness of linearity.Students will understand the multiple ways you can think about linearity and their deep connectedness.Students will understand the similarities and difference between 4 = 3x-5 and y = 3x -5.Lesson Bookmarks:Lesson 1: What do I know about linearity?Lesson 2: What is unique about linearity?Lesson 3: Tables, Graphs and EquationsLesson 4: Other situations with Tables, Graphs and EquationsLesson 5: Old Problems RevisitedLesson 6: Performance Task - Johnson Family LogoLesson 7: Will Marissa get home on time?Lesson 8: The Hippo ProblemLesson 9: On Demand Task-- The Movie RentalLesson 10: The Shoe Size Problem*Lessons 11 & 12: What story is this graph telling?*Lesson 13: Solving Systems of Equations by Graphing*Lesson 14: Solving Systems of Equations by Substitution*Lesson 15: Solving Systems of Equations Using EliminationLessons 16 & 17: Solving Systems of Equations Contextual ProblemsLesson 18: The Hamburger ProblemLesson 19: The RaceLesson 20: Graphing InequalitiesLesson 21: Graphing Systems of InequalitiesLesson 22: Solving System of Inequality Contextual ProblemsFinal Project/Reflection*Regents Questions* * - denotes new lesson in 2015 revisionStandards Addressed:A-CED.A.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.A-CED.A.3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.A-REI.C.5Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.A-REI.C.6Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.A-REI.D.10Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).A-REI.D.11Explain why the?x-coordinates of the points where the graphs of the equations?y?=?f(x) and?y?=?g(x) intersect are the solutions of the equation?f(x) =?g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where?f(x) and/or?g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*A-REI.D.12Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.F-LE.A.1.AProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.F-LE.A.1.BRecognize situations in which one quantity changes at a constant rate per unit interval relative to another.F-LE.A.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).F-LE.B.5Interpret the parameters in a linear or exponential function in terms of a context.Unit 3: Linearitytc "Beginning of Lessons" \l 2Lesson 1: What do I know about linearity?Note to Teacher: Solution Evolves to an Infinite Collection of PointsIn Unit 2 as we transitioned from equations to inequalities we discussed how the notion of solution evolves. As we went from equations to inequalities solution went from being one specific place on the number line to a region on the number line as we work with inequalities. In this unit we’ll get to see two more evolutions that were alluded to in the Note to Teacher in the last unit.In the Opening Activity in Lesson 1, on the next page, students will get to go from 5 = 2x-3 having one solution (one place on the number line) to y=2x – 3 which will transition to 2-dimensional space with an infinite collection of solutions forming a line. 5=2x-311404601257301-10234x001-10234x20434308763000y=2x-3This evolution is dramatic! Why are we now in 2-dimensions having 2 number lines? Why does a line represent the infinite collection of solutions to this equation? These are interesting questions that can be a part of the discussion when you see fit. When the time is right you may want to ask a provoking question having the students conjecture about what form they think solution can take when there are two equations and two unknowns.Teacher GuideDecision Maker: The purpose of this opening lesson is for you to find out what your students know about linearity. What do they remember and understand from what they experienced in previous mathematics classes? Students will bring with them understandings that both show some level of conceptual understanding and misunderstanding of ideas. They will often be able to recall words (e.g. slope, y-intercept, rate-of-change, etc.) or equations (e.g. y=mx+b), but these things will possibly be shared without much depth of understanding. As you listen carefully to your students ideas, be willing to be flexible with your plan because you may need to change directions. At the end of each activity do you want to bring the class together to discuss what students are thinking and helping them to get deeper about the ideas presented?Opening Activity: Learner: See yourself learning about your different students in relation to their thinking about an equation as represented with one variable and two variables. Why is this important? How does this prepare your students as they begin to think about functions? We will not be using the term “functions” until the next unit but we are beginning to work with linear functions in this unit.Artist: Mitsouka Jean Claude, a teacher at Victory Collegiate HS, made this activity into a “speak out”. She presented the two equations and question prompts to the students. After a few minutes she posed the following question: do these two equations have the same solution? Each student was given a post-it note to jot down a response with supporting evidence. She had three areas in the classroom: Yes, they are the same, No, they are different, I am not sure. Students go to the part of the room that corresponds to their idea. Then students speak out (have a discussion) about why they chose their response, eventually coming to a consensus.Look at these two equations: 5 = 2x – 3 y = 2x - 3How are they similar and how are they different? How does the meaning of solution differ for each one? In other words, what does the solution look like for each one? How many solutions does each one have?Activity 2: Learner: What do your students understand about creating a table from an equation? What meaning do they make of the table? You should be gathering information about your students so you can make sure that in this unit you are be able to support students who lack understanding of these different concepts or have misunderstandings. The other lessons in this unit will build off of this plete the following table for y = 2x – 3xy-2-10123Describe any patterns that you see. What do the patterns mean?Activity 3:Decision Maker: Since the intent is to learn about your students you want to step back as much as you can. If many students are struggling with this question then you might need to rethink what to do. Since graphing will occur throughout this unit (and throughout this course) you might find different opportunities to help students develop a real understanding of graphing. Will you need to create a lesson for the whole class on how to graph at some point in this unit? Can you have those who know what to do work together and you work with those students who are struggling?Now you are going to graph y = 2x – 3 on a coordinate plane. Can you explain what you are going to do in order to graph this equation? What do you expect the graph to look like? Why?Closing Activity: Learner: This is where you will find out a good deal of information about your students. In the 8th grade the students spent a good deal of time working with linearity and thinking about its features. Have they thought about the relationship between table, graph and equation before? Can they talk about rate of change or y intercept with meaning and understanding? All these ideas will be revisited throughout the unit, so don’t worry if their understanding is limited. Decision Maker: As a teacher you want to value all student questions. What do you want to do with these questions? Do you want to post them in the room knowing that you will answer them all throughout the unit?a) Examine the equation, table and graph. What can you say about the relationships between the three representations? Do they tell the same story? How do you know? (If you can, include slope/rate of change, and y-intercepts in your answer.)b) What do you remember from working with linearity before? What questions do you have about linearity from previous math classes or from what we explored today?Unit 3: LinearityLesson 1: What do I know about linearity?Student VersionName_______________________Date________________________Opening Activity: Look at these two equations: 5 = 2x – 3 y = 2x - 3How are they similar and how are they different? How does the meaning of solution differ for each one? In other words, what does the solution look like for each one? How many solutions does each one have?Activity 2:Complete the following table for y = 2x - 3xy-2-10123Describe any patterns that you see. What do the patterns mean?Activity 3:Now you are going to graph y = 2x – 3 on a coordinate plane. Can you explain what you are going to do in order to graph this equation? What do you expect the graph to look like? Why?Closing Activity:a) Examine the equation, table and graph. What can you say about the relationships between the three representations? Do they tell the same story? How do you know? (If you can include slope/rate of change and y-intercepts in your answer.)b) What do you remember from working with linearity before? What questions do you have about linearity from previous math classes or from what we explored today?Unit 3: LinearityLesson 2: What is unique about linearity?Teacher GuideNote to Teacher: The purpose of this lesson is to focus on the uniqueness of linear functions. We will be developing the idea of constant rate of change through the four activities. As you see we have not given the formula for slope. Can it be developed from the table---change in y over change in x? You will notice that throughout this unit we rarely introduce a formula. Formulas need to come from conceptual understanding, this way the formula makes sense because of an understanding of the concept of rate of change. You need to decide if you want to introduce a formula and what would be the best way to do it. If students don’t understand the basis of a formula it is not very useful, and they will often forget it or mix it up in their heads.Opening Activity: Learner: What do my students understand about rate of change, a concept they learned in eighth grade? A goal for this lesson is to begin to observe the idea of a common rate of change which will be further developed in Activity 2. Take a look at Activity 2 to see how language will be developed in the lesson. Decision Maker: How am I going to lead a discussion about the students’ ideas? Did I go around the room to observe how my students thought about the question? Did I observe any interesting approaches? Are there any misconceptions that I will need to raise with the class? How can I use the students’ ideas that I observed to enhance the group discussion? If students graphed each of them, which is okay why did the first and third come to be a line and not the second one? Here are three tables. The job of your group is to decide which are linear and which are not linear. Graph paper will be supplied if you want to use it. Be ready to defend your solution to the whole class.xy021528311xy00112439xy-2-10327411Activity 2: Coach: Constant Rate of change/ slope is a rich conceptual idea and you will begin to develop this concept with this activity. Students might need help in understanding what is being asked. You want them see how to move from one point to any other point. You might model the idea of moving up vertically and then moving horizontally.Decision Maker: a question you might ask is: “How would you move from one point to another on this graph? Explain the movement you would make.”Artist: The facilitation of this discussion will be very important. You want student observations about the graphs and then the tables to lead the way. How can I encourage my students to use their observations of the graphs to understand the patterns in the table?The graphs below correspond to the tables from the Opening Activity. Your job is to decide if the growth of the line is the same as you move from one point to another point on each graph. The teacher may model how you might do this. 399859510731500 476250165100045212011684000What observations can you make about the graphs?Now go back to the tables in the Opening Activity and figure out how you can determine, without graphing, if a table is or is not linear.Learner: You have spent time listening to your students and will have an idea of what your students recall and deeply understand about linearity. Listen very carefully to the words they are using. It is important for students to communicate in everyday (social) language because it not only helps students to feel relaxed but encourages them to communicate prior knowledge and new ideas freely. However, it is extremely important that students are explicitly instructed on the language of the discipline (one component of academic language). Coach: A part of our role is also to help students develop an ability to communicate in a more academic way, and to also be able to switch between our everyday language and the mathematical language at ease.Here is one idea for facilitating a discussion on the formal language of rate of change:Write down what you hear your students saying (here are a few made up phrases your students might say)I know something is linear because when I go from one point to the next it is the same.It is a pattern in the table.If the rise over run is the same it is linear.Underline, circle or write key phrases in different colorsI know something is linear because when I go from one point to the next it is the same.It is a pattern in the table.If the rise over run is the same it is linear.Ask students if they of a more academic way of saying these underlined expressions (you can give them a word bank – constant, common difference, rate of change, slope, ratio, horizontal change, vertical change)I know something is linear because when I go from one point to the next the rate of change is constant.It is a common difference in the table.If the ratio of the vertical change over the horizontal change is constant it is linear.Decision maker: How will you help students develop the formula for slope?Activity 3:Coach: This is a challenging problem. Students often want the teacher to tell them what to do. The problem with this is it stops the thinking. How can you let students grapple with it? If you need to enter into student discussion how can you ensure that you help further the thinking?Questioner: What questions might you ask to push student thinking?Does this table represent a linear relationship? Justify your answer without using a graph.xy03-3128-213-9Closing Activity/Performance Task: Rate of Change of 4Coach: This task will tell you a good deal about student understanding of rate of change. You want to intervene as little as possible. You can make it an exit ticket and use the student ideas to lead tomorrow’s lessons.The value of this problem is that it is open –ended. There is no right answer but an infinite amount of correct answers. What do your students think about that?Create a table that has a rate of change of 4Unit 3: LinearityLesson 2: What is unique about linearity?Student VersionOpening Activity: You are going to be given three tables. The job of your group is to decide which are linear and which are not linear. Graph paper will be supplied if you want to use it. Be ready to defend your solution to the whole class.xy021528311xy00112439xy-2-10327411Activity 2: The graphs below correspond to the tables from the Opening Activity. Your job is to decide if the growth of the line is the same as you move from one point to another point on each graph. The teacher may model how you might do this. 399859510731500 476250165100019151607429500What observations can you make about the graphs?Now go back to the tables and figure out how you can determine, without graphing, if a table is or is not linear.Activity 3:Does this table represent a linear relationship? Justify your answer without using a graph.xy03-3128-213-9Closing Activity: Rate of Change of 4Create a table that has a rate of change of 4Unit 3: LinearityLesson 3: Tables, Graphs and EquationsTeacher GuideNote to Teacher: This lesson is about helping students begin to see the relationship between table, graph and equation. The first activity starts with finding out what relationships students see and the last activity is a matching activity between the three representations. Opening Activity: Coach: What relationships do my students see? How do they make sense of each representation?Decision Maker: How can this inform the lesson and discussions I have with the students? Did students notice the connection between the y-intercept on the graph, table and in the equation? What are the important things you want students to know about that idea?You will be given a table with an accompanying graph and equation. Do these three represent the same linear relationship? Be ready to defend your answer to the class.46266109398000xy-2-60-22246y = 4x – 2Activity 2:Coach: This activity was set up to see what students understood in 8th grade math and what understanding your students had about the relationships that were talked about in the opening activity. What did your students notice in the Opening Activity? Did they notice the idea of rate of change in all three representations? Did they see the y-intercept in all three representations? If you feel this activity might be too early for your students you can create an activity that develops y = mx + b. But if you feel you can use the two activities to develop that idea, then go with it. How do you find an equation from a table?a) You will be given a table and you have to come up with the equation for the table…Be prepared to share your process and explanation of how you know you are correct.xy-3-60329518b) Now find an equation for this table xy439-711-11c) Describe in your words a method to find the equation when given a table.d) Find the y value for each table if x = 65.e) Using the first table, if y = 155 is the x value an integer? Justify your answer.Activity 3:Learner: Again, you want to learn how your students think in a new situation. How do they use their prior knowledge to answer a new question? How did they build off what they understood in the first two activities?How do you find an equation from a graph? Your challenge in this situation is to determine a method of figuring out the equation of the given line on a coordinate plane. Look at these two lines and try to find the equations for each line.38982659525002286063500Closing Activity/Performance Task: MatchmakerDecision Maker: How do you want to do this activity? Do you want it to be an exit ticket and you can then see what your students understand? Do you want it to be a pair activity, perhaps, homogenously grouped, to see what students together can do? What you learn from this activity should inform what you do to start the next day.Match the table, graph and equation and explain how you know that they are equivalent representations5429885195580(C)00(C)2934335147955(B)00(B)-14224052705(A)00(A)xy-2-13903xy2-25-8-26xy2-15-7-27(D) y = -2x + 2(E) y = 2x + 3(F) y= -2x +3(G)(H)(I)Unit 3: LinearityLesson 3: Tables, Graphs and EquationsStudent VersionName_______________________Date________________________Opening Activity: You will be given a table with an accompanying graph and equation. Do these three represent the same linear relationship? Be ready to defend your answer to the class.43910251905000xy-2-60-22246y = 4x – 2Activity 2:How do you find an equation from a table?a) You will be given a table and you have to come up with the equation for the table…Be prepared to share your process and explanation of how you know you are correct.xy-3-60329518b) Now find an equation for this table xy439-711-11c) Describe in your words a method to find the equation when given a table.d) Find the y value for each table if x = 65.e) Using the first table if y = 155 is the x value an integer? Justify your answer.Activity 3:How do you find an equation from a graph? Your challenge in this situation is to determine a method of figuring out the equation of the given line on a coordinate plane. Look at these two lines and try to find the equations for each line.171450444500040481255397500Closing Activity: MatchmakerMatch the table, graph and equation and explain how you know that they are equivalent representations5429885147955(C)00(C)2934335167005(B)00(B)-142240119380(A)00(A)xy-2-13903xy2-25-8-26xy2-15-7-27375348564770(H)00(H)37338009588500(D) y = -2x + 2(E) y = 2x + 3(F) y= -2x +3(G)34296352007870(I)00(I)3505200222313500Unit 3: LinearityLesson 4: Other situations with Tables, Graphs and EquationsTeacher GuideOpening Activity: Here are 2 points on a line, (-3, 4) and (2, -1) find the equation of the line. Be prepared to explain what you did and how you know you are correct.Decision Maker: There are many ways to do this. We want students to use methods that make sense to them, not methods we impose on them. Could they use y = mx + b? Could they graph it and find the y-intercept from the graph? Perhaps the most organic is using a table. If students understand that the value the y- intercept is when x = 0 then they can find the y-intercept using the patterns of the table.Activity 2:Coach: This is a pure observation activity that most students will observe. Questioner: You want your students to be able to talk about why lines with the same slope are parallel. How will you elicit that thinking?Graph each of the following equations on your calculator:y = -2x + 2y = -2x y= -2x - 3Observe the tables and the accompanying graphs. What conjectures can you make based on your observations? Do you think your conjectures will always hold true? Why?Activity 3/Performance Task: Changing the TableDecision Maker: How nuanced is your student’s understanding of rate of change? What supports might they need for this problem? Do you want to have pairs work with this problem? If this seems too challenging for some of your students do you want to give certain groups an easier table to work with?Look at the following table. Your job is to make as few changes as possible to the table so that the rate of change now becomes -3. Justify your answer.xy-2-13-110-5Unit 3: LinearityLesson 4: Other situations with Tables, Graphs and EquationsStudent VersionName_______________________Date________________________Opening Activity: Here are 2 points on a line, (-3, 4) and (2, -1) determine the equation of the line. Be prepared to explain what you did and how you know you are correct.Activity 2:Graph each of the following equations on your calculator:y = -2x + 2y = -2x y= -2x - 3Observe the tables and the accompanying graphs. What conjectures can you make based on your observations? Do you think your conjectures will always hold true? Why?Activity 3/Performance Task: Changing the TableLook at the following table. Your job is to make as few changes as possible to the table so that the rate of change now becomes -3. Justify your answer.xy-2-13-110-5Unit 3: LinearityLesson 5: Old Problems RevisitedTeacher GuideNote to Teacher: The purpose of this lesson and the upcoming lessons is to see how students think about linearity in contextual situations. How can they use the concepts and procedures that were developed in the first set of lessons to help them in working with the problems? We will begin with revisiting problems that you probably used in the first unit. If the problems are new for the students, that will be fine also.Opening Activity: Decision Maker: How do you want to facilitate this lesson? What can you learn about your students from having them revisit these two problems? How can that inform the decisions you are going to make?Coach: You want students to feel free to try any approach. You also want to see if they can use their understanding of linearity to solve the problem. How will they use rate of change? Will they create an expression or an equation to help solve these two problems? How can you help your students see the value of an algebraic approach to solving these problems?Decision Maker: Will you provide your students with a KNN chart, or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext StepsWe worked on Table Hopping in Unit 1. We are now going to revisit this problem since you now have a deeper understanding of linearity. How might you now solve the problem using what you have learned about tables and equations?46856657747000Table HoppingThe school is having a big event in the gym and wants to use their rectangular tables. 6 people can sit together at 1 rectangular table. If 2 tables are placed together, 10 people can sit together. 14 people can sit if three tables are placed together. If the pattern continues…..How many people could sit if 100 tables were put together? Explain your thinking. Would your pattern make it possible to seat 150 people without any empty seats? Justify your answer mathematically.Activity 2:We worked on Crossing the River in the first unit. How might you now solve the problem using tables and equations?Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps5721350-26733500Crossing the River ProblemEight adults and two children need to cross the river and they only have one boat available. The boat can only hold one adult or one child or two children. Everyone in the group is able to row the boat.How many one-way trips will it take for everyone to cross the river?Unit 3: LinearityLesson 5: Old Problems RevisitedStudent VersionWe worked on Table Hopping in Unit 1. We are now going to revisit this problem since you now have a deeper understanding of linearity. How might you now solve the problem using what you have learned about tables and equations?46856657747000Table HoppingThe school is having a big event in the gym and wants to use their rectangular tables. 6 people can sit together at 1 rectangular table. If 2 tables are placed together, 10 people can sit together. 14 people can sit if three tables are placed together. If the pattern continues…How many people could sit if 100 tables were put together? Explain your thinking. Would your pattern make it possible to seat 150 people without any empty seats? Justify your answer mathematically.Activity 2:We worked on Crossing the River in the first unit. How might you now solve the problem using tables and equations?5721350-26733500Crossing the River ProblemEight adults and two children need to cross the river and they only have one boat available. The boat can only hold one adult or one child or two children. Everyone in the group is able to row the boat.How many one-way trips will it take for everyone to cross the river?Unit 3: LinearityLesson 6: Johnson Family LogoTeacher GuideNote to Teacher: This is a group performance task. Place students into groups of two or three. Do you want to group them homogenously or heterogeneously?Decision Maker: Will you provide your students with a KNN chart or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps179070012763500Opening Activity: The ohnson Family LogoDecision Maker: This is a full activity for the whole period. Do you want to let students work the whole period or do you want to bring the class together for the last 10-15 minutes? What would be most valuable for the students? Where would you learn the most about your student thinking?Coach: Should I keep away from the students while they work or should I enter in when necessary? If I enter in why would it be helpful? How do I support students without showing them what to do? What questions might I ask in this situation?Figure 1Figure 2Figure 3Jackie, Jasmine and Jeremy, brothers and sisters in the Johnson family, have presented you with a problem. The three figures above are the basis for the design of their logo. They plan to put this logo on stationery, on their jackets and even on different sized buildings. Your task is to determine the answers to certain questions that came up in their planning-95250146050001) Draw the fourth figure.-762006985002) How many squares would they need for the 30th figure? Show your work.-7620086995003) Many students thought answering question 2 took too much time. Describe an efficient method for calculating the number of squares for any size figure. Explain why this method would work.-66675-2540004) If we have 332 squares, could we make a design that uses all of them? If not, how many are left over? Explain your mathematical thinking.-10477553340005) If you were asked to draw the 0th figure what would it look like? What would this mean in terms of the generalization?-9525066675006) If we want to create rectangles from this design, how many squares need to be added to each figure? Create a mathematical rule to describe this new requirement.Unit 3: Linearity - Lesson 6Student Version184785012763500The ohnson Family LogoFigure 1Figure 2Figure 3Jackie, Jasmine and Jeremy, brothers and sisters in the Johnson family, have presented you with a problem. The three figures above are the basis for the design of their logo. They plan to put this logo on stationery, on their jackets and even on different sized buildings. Your task is to determine the answers to certain questions that came up in their planning-152400154940001) Draw the fourth figure.-123825-10160002) How many squares would they need for the 30th figure? Show your work.-114300-2540003) Many students thought answering question 2 took too much time. Describe an efficient method for calculating the number of squares for any size figure? Explain why this method would work.476258890004) If we have 332 squares, could we make a design that uses all of them? If not, how many are left over? Explain your mathematical thinking.-95250173355005) If you were asked to draw the 0th figure what would it look like? What would this mean in terms of the generalization?046990006) If we want to create rectangles from this design, how many squares need to be added to each figure? Create a mathematical rule to describe this new requirement.Unit3: LinearityLesson 7: Will Marissa get home on time?Teacher GuideNote to Teacher: This problem is a little different from the previous problems - you are only given two data points. How do you use that information to try to solve this problem? Can this problem be solved purely by using a table? Can an equation be create and use to solve the problem? Can it be solved graphically? How do students work with the negative rate of change? Again try to solve it in multiple ways. It will give you useful information when you teach the lesson.Opening Activity: Coach: How do you want to facilitate this lesson? An interesting approach might be putting students in groups and each group has to solve it in a particular way. Once that has been completed you form groups of three where each member solved it differently. They need to show each other how they solved it then the groups have to then do a write up about the different approaches and how they see they are related. (This is called a Jigsaw activity.)Decision Maker: Will you provide your students with a KNN chart will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps60293258509000Will Marissa Get Home on Time?Marissa went to a party at her friend’s house. She promised her strict father that she would get home by 11PM. He told her that she would be grounded for a month if she comes home one second past 11. Was she grounded?Here is some information to help you figure this out. She left the party at 9:50 P.M.She drove an old Buick that made a lot of noise as she drove itMarissa lived 18 miles from where the party was heldShe drove at the same rate the whole way homeMarissa was 14 miles from home after16 minutesWas she grounded? How do you know?Unit3: Linearity - Lesson 7Student Version60198005524500Will Marissa Get Home on Time?Marissa went to a party at her friend’s house. She promised her strict father that she would get home by 11PM. He told her that she would be grounded for a month if she comes home one second past 11. Was she grounded?Here is some information to help you figure this out. She left the party at 9:50 P.M.She drove an old Buick that made a lot of noise as she drove itMarissa lived 18 miles from where the party was heldShe drove at the same rate the whole way homeMarissa was 14 miles from home after16 minutesWas she grounded? How do you know? Unit 3: LinearityLesson 8: The Hippo ProblemTeacher GuideNote to Teacher: As we work on these linear function problems we hope that students will become more independent in solving the problems. Remind them to use what they learned from the other problems to help them to think about this problem. It will be important for students to think about how this problem is similar and different from the previous problems. In this situation we are working with a positive slope that passes through quadrants I and IV. Make sure you have spent a good deal of time with the problem yourself. Think about different approaches to the problem. How will you group the students for this problem?Decision Maker: This problem should take a full period. You can use this problem as an open-ended task and let students solve it any way they can. If necessary, you can recommend to the class that they can solve it in any way they choose. They can create a table, equation and/ or graph the points on a coordinate plane so that can help them visualize what is going on. If you need to enter into the process more deeply with some students that is okay, but ask them questions, don’t just tell them what to do?. What questions might you ask students who are struggling with this problem? Artist: How can you best facilitate the discussion that shows different approaches to solving the problem? How can you help students see the connections between different approaches? Which approach makes the best sense to you? Why?Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps52558953556000The Hippopotamus ProblemIn order to hunt hippopotamuses, a hunter must have a hippopotamus-hunting license. Since Hakim the Hunter can sell the hippos he catches, he can use the proceeds to pay for part of or the entire license. If Hakim only catches three hippos, he is still $2050 in debt. If he catches seven hippos, he makes a profit of $1550. The African Game and Wildlife Commission allows a limit of ten hippos per hunter. -10477516764000What is the maximum amount of money Hakim can make? What is your evidence?-114300000What is the cost of the license? How do you know?-114300-63500What is the contextual meaning of the rate of change, y-intercept and x-intercept?Unit 3: Linearity - Lesson 8Student Version56197503238500The Hippopotamus ProblemIn order to hunt hippopotamuses, a hunter must have a hippopotamus-hunting license. Since Hakim the Hunter can sell the hippos he catches, he can use the proceeds to pay for part of, or the entire license. If Hakim catches only three hippos, he is still $2050 in debt. If he catches seven hippos, he makes a profit of $1550. The African Game and Wildlife Commission allows a limit of ten hippos per hunter. -2857517653000What is the maximum amount of money Hakim can make? What is your evidence?-28575-1968500What is the cost of the license? How do you know?-142875-63500What is the contextual meaning of the rate of change, y-intercept and x-intercept?Unit 3: LinearityLesson 9: On Demand Task-- The Movie RentalTeacher GuideNote to Teacher: This task should be given to students as an on-demand task. That means they should work alone or in groups of two and try to solve this problem in any way they can. The numbers in this problem make it a little messy. What do students do with that? Can they create an equation to solve this problem (the most efficient way)?Decision Maker: Since the numbers are messy you might want to simplify the numbers for some students. Thus giving the same task with different numbers to students might make sense. The goal is to see what students do with this problem.Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps48958506223000The Movie Rental Keisha has a movie rental card worth $ 175. After she rents the first movie the card’s value is $ 172.25. After she rents the second movie its value is $ 169.50. After she rents the third movie the card is worth $166.75. Assume the pattern continues. Keisha rents a movie every Friday. -18288017780000How many weeks in a row can she afford to rent a movie, using the rental card only? Explain how you arrived at your answer. Use as many representations (table, graph, equation) as you can in solving this problem.Suppose the rental plan goes up $ 0.25 for renting a movie. How will this affect the number of weeks Keisha can rent a movie?Unit 3: Linearity - Lesson 9: On Demand Task50673007048500Student VersionThe Movie Rental Keisha has a movie rental card worth $ 175. After she rents the first movie the card’s value is $ 172.25. After she rents the second movie its value is $ 169.50. After she rents the third movie the card is worth $166.75. Assume the pattern continues. Keisha rents a movie every Friday. -12382513906500How many weeks in a row can she afford to rent a movie, using the rental card only. Explain how you arrived at your answer. Use as many representations (table, graph, equation) as you can in solving this problem.-1047752730500Suppose the rental plan goes up $ 0.25 for renting a movie. How will this affect the number of weeks Keisha can rent a movie?Unit 3: LinearityLesson 10: The Shoe Size ProblemTeacher Guide62972951206500Note to Teacher: All of the problems in this unit so far have presented situations that are perfectly linear – that is the data forms a perfectly straight line. Now we will give students an experience with data that approximates a line and can therefore be modeled by a linear equation. Here students will learn to use technology (graphing calculator, Excel spreadsheet, calculator app) to perform a linear regression. We also want students to see the value of mathematics as a descriptor of observations of the world around us, and that these descriptive models can help us to predict future behaviors.Opening Activity: Shaquille O’Neal is seven feet one inch (7’ 1”) tall and has a fifteen inch foot. What size shoe do you think he wears? How did you come up with your conjecture? Activity 2: Learner: Students learned about scatter plots and line of best fit in eighth grade, so it shouldn’t be an entirely new idea to them. Are students confused by what to do, or do they simply connect the points? Ask students how this graph is different from other graphs you’ve worked with in this unit. Do they come to the conclusion that they need to construct a straight line?Decision maker: What units will you use for the foot length? What variable (length of foot or shoe size) should go on the x-axis? the y-axis?Decision maker: How do you want to facilitate the data collection? We ask you to collect four data points, but do you want to collect more points – perhaps one male from each group, or all males in the class? Do you want to use a female, say – Brittney Griner or Serena Williams – and work on creating two different graphs? Is it better for the whole class to share the same data so that they can share their graphs in a more meaningful way? The size shoe a person needs varies linearly with the length of his or her foot. Use this relationship to determine a way to predict the shoe size of any man.Gather data (shoe size and the foot length) for at least four young men in the class. Graph the data. What do you observe? Based on the points you graphed, do you agree that there is a linear relationship between shoe size and foot length?How do you think you can use this graph to determine Shaquille O’Neal’s shoe size? Explain.Activity 3: Learner/Coach: Encourage students to explain how they drew their line and why their lines may have been different from other students. Students at this point in the unit should be comfortable with finding the slope and y-intercept of a graph and using it to write the equation of a line in slope-intercept form. Coach: A new take-away you want students to have for now is that a straight line can be used to generalize an observed behavior, and we call that modeling. We will be exploring other types of models in the units ahead. Another take away we would like for students is that there is A LINE that best represents their data and a beginning sense of what makes one line better than another when modeling data.Decision Maker: How will you facilitate a share out? One idea is to have students post their graphs on chart paper and discuss how they went about determining the best fit line.We already know that the graph should be linear. On your own, plot the points and create the line of best fit (the straight line that best models the data in your graph). Describe how you did pare the line with the lines of other students in your group. Your task here is to have a group discussion as to why your line better fits the data than the others in your group. You will come to a consensus (agree as a group) for your group’s line of best fit. Be prepared to share your group’s graph with the class and how you determined it was the best from your group.Write the equation of the line of best fit that you created. Explain what the different parts of your equation represent. Activity 4: We can use technology to easily find the line of best fit. [We will look more closely at the mathematics behind this later in the year when we explore Statistics.] Note to Teacher: Here are links to the calculator toolkit instructions for creating a table of data* and performing linear regression** on TI-83 and TI-84 plus calculators.* ** to Teacher: Students might want to know why one line makes the most sense for this data. We will delve deeper into how to more accurately model data in the Statistics unit later in the year. Decision Maker: How will you facilitate this technology lesson? Perhaps you can have students create a calculator toolkit in their notebook.What did the linear regression determine to be the line of best fit? Explain what each part of the equation represents.Provocateur: Can a person have a 0 inch foot? Do negative shoe sizes exist? This is an example of how the y-intercept (and x-intercept) can sometimes be meaningless depending on the context of the problem. Do all values of x for the line of best fit make sense? Which ones do or which don’t’? Explain why.Closing Activity: Provocateur: If your groups didn’t use the same data an interesting conversation would be to discuss what caused the differences in the shoe size for Shaquille O’Neal.Decision Maker: Shaq’s shoe size is 22. How do you want to share this information? What type of discussion do you want to have? Some questions to consider are: How close did you come to that number? What accounts for the difference between your model and his shoe size?Use your equation or graph to determine Shaquille O’Neal’s shoe size. Exit Slip: Decision Maker: If you chose to do a woman’s foot, of course you will change this question.What would be the shoe size of a male with a nine-inch foot? Explain how you would use the equation or graph to determine this.Extension: Does a woman’s shoe size follow the same pattern? Test it out with some young women in your class.Unit 3: LinearityLesson 10: The Shoe Size ProblemStudent Version5373370254000Opening Activity: Shaquille O’Neal is seven feet one inch (7’ 1”) tall and has a fifteen inch foot. What size shoe do you think he wears? How did you come up with your conjecture? Activity 2: The size shoe a person needs varies linearly with the length of his or her foot. Use this relationship to determine a way to predict the shoe size of any man.Gather data (shoe size and the foot length) for at least four young men in the class. Graph the data. What do you observe? Based on the points you graphed, do you agree that there is a linear relationship between shoe size and foot length?How do you think you can use this graph to determine Shaquille O’Neal’s shoe size? Explain.Activity 3: We already know that the graph should be linear. On your own, plot the points and create the line of best fit (the straight line that best models the data in your graph). Describe how you did pare the line with the lines of other students in your group. Your task here is to have a group discussion as to why your line better fits the data than the others in your group. You will come to a consensus (agree as a group) for your group’s line of best fit. Be prepared to share your group’s graph with the class and how you determined it was the best from your group.Write the equation of the line of best fit that you created. Explain what the different parts of your equation represent. Activity 4: We can use technology to easily find the line of best fit. [We will look more closely at the mathematics behind this later in the year when we explore Statistics.] What did the linear regression determine to be the line of best fit? Explain what each part of the equation represents.Closing Activity: Use your equation or graph to determine Shaquille O’Neal’s shoe size. Exit Slip: What would be the shoe size of a male with a nine-inch foot? Explain how you would use the equation or graph to determine this.Extension: Does a woman’s shoe size follow the same pattern? Test it out with some young women in your class.Unit 3: LinearityLessons 11 & 12: What story is this graph telling?Teacher Guide Note to Teacher: Story graphs are a good way to help students make sense out of what is represented in a graph. In their sense making process we want students to connect rate of change and other features to the real world event. For instance, what does (0, 0) mean in terms of the event? What do different rates of change mean within the event? [All of these are piecewise functions, which we will re-visit at the end of unit 6.]Decision Maker/Artist: Could acting these out help students make the connection from the words to the graph? How can technology help collect data of an experience the students have so the students can connect their experience to different graphs? Opening Activity:Learner: Notice how we move away from numbers so that we can focus on students’ conceptual understanding of rate of change in terms of increasing and zero slope, and how it fits into a context. In asking students to graph the other two situations, we are trying to learn what students understand about comparing rates of change graphically. It’s okay if students share incorrect graphs. You are just getting a better sense of their understanding. In the next few lessons students will gain more experience with similar graphs. Decision Maker: If you’d like, you can have them re-visit this opening activity at the end of the lesson to ask them if they would change they’re graphs. It could give you a clearer sense of the knowledge they gained through the activities.Here are descriptions of three runners in a race. Alex got off to a slow start, then ran a little faster, and then ran even faster as he finished the race.Manny started off running quickly, but then tired and slowed down. He slowed down even more as he neared the end of the race.Rajib started quickly, stopped to tie his shoe, and then ran even faster than before.Which runner’s graph is shown below? 59055011049000Explain how the features of the graph helped you decide.322897523241000left20955000Draw a graph that represents each of the other two situations. Explain the difference between each of the two graphs.Activity 2: Matching GraphsLearner: In these examples, students there is a little more nuance in comparing rates of change without values. Can students do this visually? Questioner: Students will struggle with the flat and negative slopes on the graph. Some questions you can students: What relationship is the graph representing? How many different parts are represented in the graph? As time is increasing, what is happening to the distance in the graph? What is the same about Graphs B and C? What is the different? Here are five stories and six graphs. Match the five stories with their graphs. 45243751682115Graph F00Graph F24384001739265Graph E00Graph E3048001672590Graph D00Graph D304800100965Graph A00Graph A241935091440Graph B00Graph B452437591440Graph C00Graph C Story 1:After the party, Alec walked slowly home.Story 2: Karlo ran from his home to the bus stop and waited. He realized he had missed the bus so he walsked home.Story 3: Gilda walked slowly along the road, stopped to look at her watch, realized she was late, and then started running.Story 4: Fatima walked to the store at the end of her block, bought a coffee, and then ran all the way back home.Story 5: Barry went for a walk with some friends. He suddenly realized he left his cellphone behind. He ran home to get it and then had to run to catch up with the others.One of the graphs couldn’t possibly have a story. Determine which graph it is, then write about it what makes it unique.Activity 3: Learner: Are students able to choose appropriate labels for the axes? Are students able to distinguish between contexts: the first graph could represent distance from starting point over time while the second graph could represent total distance travelled over time. It is important for students to be able to develop a sense of different contexts as well as the language to describe those contexts.Kasim enjoys swimming laps at the community pool. He jumps into the swimming pool and swims to the other end of the pool. He rests for a while, then swims back.Examine the two graphs below. Can both graphs be used to represent this story?Title:7092952413000Title:8235959969500If you think, ‘Yes, both graphs can describe the story.’:Label each graph with appropriate axes (units)Give a title to each graph that explains what the graph is about.If you think, ‘No, only one of the graphs describes the story.’:Label the correct graph with appropriate axes (units)Give a title to the graph that explains what the graph is about.Closing Activity:Below are descriptions and graphs of the two other runners in a race from the Opening Activity. Examine the graphs and match the correct description to each graph. Explain how the comparing rate of change helped you make each choice.Alex got off to a slow start, then ran a little faster, and then ran even faster as he finished the race.Manny started off running quickly, but then tired and slowed down. He slowed down even more as he neared the end of the race.75565889000Opening Activity: Note to Teacher: Grace O’Keeffe, a teacher at Hudson H.S., has done a lot of work with her students around story graphs, and uses different units and types of functions throughout the year. Here is a sample of her work. This would be one of the first story graphs her students would encounter for the year. You can contact Grace on Yammer with any questions. Coach: You want to emphasize to students that here we are working with time and speed as opposed to time and distance. Notice how when the units are changed, a zero slope takes on a different meaning. It is actually more intuitive for students to think that the zero point on the graph (point E) represents no movement, but it is not as natural to think that a flat line (points A to B) represents movement at a constant speed. When students struggle with this, ask them what is happening at each section of the graph (A-B, the speed is not changing, but this means a constant speed).Provocateur: How does the story change if instead of speed vs. time this was distance vs. time?Carmen and Marta are on their way to school. Examine the graph below, which represents their walk to school. Answer the following questions about the graph. -7620-190500For what interval(s) of time are Carlos and Maria stopped? For what interval(s) of time are they walking at a constant speed?Which of intervals of time are they increasing speed at the same rate? How do you know?Explain what is happening in interval D-E. Activity 2:Learner: Here is a piecewise-graph with numbers but no units. Are students able to interpret rate of change on a graph? A graph, consisting of four parts, is shown below. The graph represents a relationship between x and y on the interval [0, 10].-5715-12065001. For what interval of x does the graph have a rate of change of 2? Justify your answer.2. For what interval on the graph is the rate of change zero? Explain your answer.3. Which interval has the greatest rate of change? Explain your reasoning.4. Write one new statement about this graph.Activity 3:Note to Teacher: This question appeared on the August 2015 CC Algebra I Regents exam. Learner: Do students understand “constant speed of 60 mph for 2 hrs.” and how to graph that? How do they make sense of the information given for different intervals of time? Are they able to represent speed as the rate of distance changing over time? Do they understand that on this graph, standing still means the rate of change is zero and there is no change in distance?Decision Maker: How do you want to work with the graphs the students make in response to the story? How do you want to discuss any discrepancies? Looking at student errors/discrepancies provide a great learning opportunity.446405889000Activity 4:Note to Teacher: This was item #2 on the June 2015 CC Algebra I Regents exam. It appeared as a multiple choice question, but we felt it was more interesting as an open-ended question.Learner: What can you learn about your students’ understanding of rate of change through this problem? You might want to ask students specific questions about the graph, such as: which interval was the jogger increasing his speed the fastest?Decision maker/Artist: as a scaffold you might want to give select students phrases that they would have to match with parts of the graph (The jogger was increasing speed for the first two minutes.; The runner ran 3 miles per hour for three minutes.; etc.). If a student is struggling, could he/she make comments about particular portions of the run?left17145000Use the graph to tell a story about the jogger’s trip. (Hint: the jogger’s rate of change can help you describe the graph.)Provocateur: Did the runner get back to where he/she started?Decision Maker: Have a discussion about the differences in the stories and the misconceptions that arise in the different stories. Closing Activity: Examine this graph and record your observations. left1016000Would you interpret How would you compare the rate of change of intervals 0 – 2 and 2 – 4? How about 4 – 5 and 5 – 7? Do you have any questions?Learner: These questions will challenge your students’ conception of rate of change. How do they make sense of the rate of change 0 – 2 and 2 – 4? Does not having a context trip them up?If this was a speed versus time graph, what would be the meaning of the interval 2 – 4? Would the meaning change if the graph represented distance versus time?Note to Teacher: We want students to recognize the meaning magnitude of rate of change takes on depending on the context. How do you compare equal but opposite rate of change? When comparing rates of change do we rely solely on the magnitude or is the sign also considered? Performance Task: Story GraphsCreate your own story graph. Explain the story using parts of the graph. Use activities four and five as examples.Unit 3: LinearityLessons 11 & 12: What story is this graph telling?Student VersionName_______________________Date________________________Opening Activity:Here are descriptions of three runners in a race. Alex got off to a slow start, then ran a little faster, and then ran even faster as he finished the race.Manny started off running quickly, but then tired and slowed down. He slowed down even more as he neared the end of the race.Rajib started quickly, stopped to tie his shoe, and then ran even faster than before.Which runner’s graph is shown below? 59055011049000Explain how the features of the graph helped you decide.322897523241000left20955000Draw a graph that represents each of the other two situations. Explain the difference between each of the two graphs.Activity 2: Matching GraphsHere are five stories and six graphs. Match the five stories with their graphs. 45243751682115Graph F00Graph F24384001739265Graph E00Graph E3048001672590Graph D00Graph D304800100965Graph A00Graph A241935091440Graph B00Graph B452437591440Graph C00Graph C Story 1:After the party, Alec walked slowly home.Story 2: Karlo ran from his home to the bus stop and waited. He realized he had missed the bus so he walsked home.Story 3: Gilda walked slowly along the road, stopped to look at her watch, realized she was late, and then started running.Story 4: Fatima walked to the store at the end of her block, bought a coffee, and then ran all the way back home.Story 5: Barry went for a walk with some friends. He suddenly realized he left his cellphone behind. He ran home to get it and then had to run to catch up with the others.One of the graphs couldn’t possibly have a story. Determine which graph it is, then write about it what makes it unique.Activity 3: Kasim enjoys swimming laps at the community pool. He jumps into the swimming pool and swims to the other end of the pool. He rests for a while, then swims back.Examine the two graphs below. Can both graphs be used to represent this story?Title:7092952413000Title:8235959969500If you think, ‘Yes, both graphs can describe the story.’:Label each graph with appropriate axes (units)Give a title to each graph that explains what the graph is about.If you think, ‘No, only one of the graphs describes the story.’:Label the correct graph with appropriate axes (units)Give a title to the graph that explains what the graph is about.Closing Activity:Below are descriptions and graphs of the two other runners in a race from the Opening Activity. Examine the graphs and match the correct description to each graph. Explain how the comparing rate of change helped you make each choice.Alex got off to a slow start, then ran a little faster, and then ran even faster as he finished the race.Manny started off running quickly, but then tired and slowed down. He slowed down even more as he neared the end of the race.75565889000Opening Activity: Carmen and Marta are on their way to school. Examine the graph below, which represents their walk to school. Answer the following questions about the graph. -7620-190500For what interval(s) of time are Carlos and Maria stopped? For what interval(s) of time are they walking at a constant speed?Which of intervals of time are they increasing speed at the same rate? How do you know?Explain what is happening in interval D-E. Activity 2:A graph, consisting of four parts, is shown below. The graph represents a relationship between x and y on the interval [0, 10].-5715-12065001. For what interval of x does the graph have a rate of change of 2? Justify your answer.2. For what interval on the graph is the rate of change zero? Explain your answer.3. Which interval has the greatest rate of change? Explain your reasoning.4. Write one new statement about this graph.Activity 3:446405889000Activity 4:left17145000Use the graph to tell a story about the jogger’s trip. (Hint: the jogger’s rate of change can help you describe the graph.)Closing Activity: Examine this graph and record your observations. left1016000Would you interpret How would you compare the rate of change of intervals 0 – 2 and 2 – 4? How about 4 – 5 and 5 – 7? Do you have any questions?If this was a speed versus time graph, what would be the meaning of the interval 2 – 4? Would the meaning change if the graph represented distance versus time?Performance Task:Create your own story graph. Explain the story using parts of the graph. Use activities four and five as examples.Unit 3: LinearityLesson 13: Solving Systems of Equations by Graphing Teacher GuideNote to Teacher: Previous versions of this unit combined graphing and substitution into one lesson. They have now been separated into two lessons and several changed have been made to each.Note to Teacher: What does solution look like with two lines?When we work with two equations with two variables solution takes on a new look. Before it was dependent on the relationship of a variable or between variables, now solution is dependent upon the relationship between the two equations. This evolution leads us to having some new options:1-SolutionIn the Opening Activity the students are presented with y = 3x - 2 and y = x + 4, which leads us to a new and interesting observation of solution. Taken separately each relationship between x and y provides us with an infinite set of points that for two distinct lines. In the case of these two equations however they share one specific place in space. Thus we have two equations with two unknowns and we have one solution, one place in space. Why is this similar to when we have one equation with one unknown? Do we have a potential pattern beginning to form in our understanding of solution? Previously we had one equation with one unknown and had one place in space (1-D space) that represented solution. Now we have two equations with two unknowns and we have one place in space (2-D space) that represents solution. Provocateur: As we think about these interesting ideas with students we can provoke them beyond this class and even traditional high school course sequencing with a question like: What does this mean for 3 equations with 3 unknowns? What about 4 equations and 4 unknowns? Then, for those students really looking for a challenge, what would it look like to demonstrate this?Situations where there are infinite solutions or no solutions are built into the lessons found in this unit. In keeping with our thinking about solution you may find it interesting to take up a couple of provoking questions with your students that begins the conversation about these two unique situations.Infinite Solutions:The case of an infinite number of solutions is fascinating because it relates back to the idea of creating equivalent equations, which we worked with when solving in Unit 2. For instance, if we have the system y=3x+8 and -6x+2y=16, we see that the second equation can be created by multiplying the first by 2 and rearranging it. Thus, if we alter these two equations in the same way they will remain equivalent. Therefore, this system has an infinite number of solutions because the equations are equivalent. This infinite collection of points that creates the line below will satisfy the relationship between x and y in both equations.Though in a more sophisticated way, we’ll also look at this idea in Lesson 13, Activity 5.No Solution:Systems of linear equations that that produce no solution exhibit a similar characteristic that can give rise to an interesting discussion. If we have the system y = 3x + 2 and y = 3x -2 there is no point in the infinite Cartesian plane that will satisfy both equations for the same x value. What causes these two equations to never share a common point? How could one of the equations be changed so that there is a solution? How could one of the equations be changed so that there are an infinite number of solutions? How do these alterations affect the table or graph?These ideas are so interesting to consider with students and can open up a creative exploration for students.Opening Activity:Coach: This opening activity is written to see what students observe about a system of equations. Will they talk about the point of intersection? What does that point mean? How would you know if you did it correctly? What would the term solution mean in a system of equations? How does it differ from the way we talked about solution before?3019425158115xy00xyGraph these two equations. y = 3x - 2 and y = x + 4013335000Write down your observations.What can be said about the system of equations y = 3x - 2 and y = x + 4 and its graph?Note to teacher: At some point you should introduce students to the term system of equations. Activity 2: What do you think will happen when we graph these two equations? y = 3x + 5y = 3x -5Graph them. What do you notice about the systems of equations y = 3x + 5 and y = 3x -5 and its graph?What hypothesis do you have about this type of system of equations? What name would you give this kind of system? Explain.Note to Teacher: Students will be able to see from the graph that the system has no solutions. We are also trying to get students to see that lines with the same slope and different y-intercepts are parallel. In order for students to make that connection, it is necessary for them to recall that the coefficient of the x in the linear equation is the slope. Facilitate a discussion around these connections.Decision Maker: We included problems where students need to rearrange the equation to y = mx + b. What is the value in having students work with different types of equations? Learner: Do students recognize these as equivalent equations, which they just learned in Unit 2?Activity 3: Graph these two equations: y = 3x + 2 and 2y – 6x = 4. What can you say about the solution to this system of equations y = 3x + 2 and 2y – 6x = 4 and its graph?What hypothesis do you have about this type of system of equations? What name would you give this kind of system? Explain.Decision Maker: Students will be able to see that these are the same equation in different form. What does that say about the solution? What connection does this have to equivalence? What do you want to take up in the discussion with students?Activity 4: Based on the first three activities, determine if each system of equations will have one solution, no solutions or infinitely many solutions, without graphing. Explain how you know.1) y=4x+103y- 12x= 92) y=2x+3y= -2x+5Activity 5: We determined that the solution to y=3x-2y=x+4 is (3,7). How can we be sure, without graphing, that (3,7) is a solution? Explain.Exit Slip: What is the solution to the system of equations graphed below? How would you explain to someone why that’s the solution?6616702540000What is the solution to the system of equations represented in the tables below? Why does your solution make sense?y = 2x + 8XY-5-2-40-32-24-1608110212314416518y = 5x - 1XY-5-26-4-21-3-16-2-11-1-60-11429314419524Homework: Find the solution to each system of equations graphically.2x-3y=182x+y=2 y- x=3y=-4x-2 left1333500Unit 3: LinearityLesson 13: Solving systems of equations by graphing.Student VersionName_______________________Date________________________Opening Activity: Graph these two equations. y = 3x - 2 and y = x + 476200076200xy00xyWrite down your observations.What can be said about the system of equations y = 3x - 2 and y = x + 4 and its graph?Activity 2: What do you think will happen when we graph these two equations? y = 3x + 5y = 3x -5Graph them. What do you notice about the systems of equations y = 3x + 5 and y = 3x -5 and its graph?What hypothesis do you have about this type of system of equations? What name would you give this kind of system? Explain.Activity 3: Graph these two equations: y = 3x + 2 and 2y – 6x = 4. What can you say about the solution to this system of equations y = 3x + 2 and 2y – 6x = 4 and its graph?What hypothesis do you have about this type of system of equations? What name would you give this kind of system? Explain.Activity 4: Based on the first three activities, determine if each system of equations will have one solution, no solutions or infinitely many solutions, without graphing. Explain how you know.1) y=4x+103y- 12x= 92) y=2x+3y= -2x+5Activity 5: We determined that the solution to y=3x-2y=x+4 is (3,7). How can we be sure, without graphing, that (3,7) is a solution? Explain.Exit Slip: What is the solution to the system of equations graphed below? How would you explain to someone why that’s the solution?6616702540000What is the solution to the system of equations represented in the tables below? Why does your solution make sense?y = 2x + 8XY-5-2-40-32-24-1608110212314416518y = 5x - 1XY-5-26-4-21-3-16-2-11-1-60-11429314419524Homework: Find the solution to each system of equations graphically.2x-3y=182x+y=2 y- x=3y=-4x-2 left1333500Unit 3: LinearityLesson 14: Solving Systems of Equations by Substitution Teacher GuideLearner: In this activity students explore the concept of equivalence (and variables). We recommend you spend some time yourself playing around the different ways one can navigate their way through the problem. Allow your experience with the problem to help inform how you answer facilitate this with your class. Learner: Are your students able to recognize that equal quantities can be used interchangeably? Do some students represent the relationships algebraically? How do students reason with the task? Decision Maker: Do you want to give these relationships on slips of paper? One approach we’ve taken when using this activity in workshops is to give each line one on a slip of paper; as pictured below. This way the relationships can be physically manipulated and allow a variety of different ways to determine which team won the tug of war.60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 437515046037500386334048577500276415535941000 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 Decision Maker/Provocateur/Learner: How do you want to facilitate the follow up discussion? In ways can honoring the different approaches that students took to figure out who won the tug war help us talk about equivalence? The focus of this follow up discussion is more on the process and usage of equivalence than which team wins the tug of war.Here are some different ways we’ve seen folks work their way through the Tug of War:Physically replace the 4 oxen in the last line with 5 horses from the first line and then physically replacing the elephant in the last line with the oxen and 2 horses from the second line. Making the last line 5 horses and an ox versus 5 horses. 1 ox is 5/4 of horses and then turned everything into horses. This means an elephant is 2 and 5/4 horses and so on.Others have made substitutions so that the last line would be 5 oxen versus 4 oxenOpening Activity: Tug of WarFour oxen are as strong as five horses.60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 An elephant is as strong as one ox and two horses437515046037500386334048577500276415535941000 Who will win the tug-of-war pictured below? 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 Note to Teacher: If you want to use the slips print and cut sets out.60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 437515046037500386334048577500276415535941000 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 437515046037500386334048577500276415535941000 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 437515046037500386334048577500276415535941000 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 Activity 2: Decision Maker: Let students grapple with this problem for a while. Then you should have a class discussion about this algebraic method of solving a system of equations. Even though the ‘and’ is not stated explicitly, it is implied when given a system of equations. Problems written using this notation are asking one to find the solution(s) where y=3x-2 and x+4=y.Questioner: This might be a good question to ask: Since both 3x – 2 and x + 4 both equal to y what can you say about 3x – 2 and x + 4?Coach: If students are having a hard time creating one equation from the two equations using substitution, then refer back to the Tug of War and consider how that experience might help us here.Using the same two equations from yesterday’s Opening Activity, how could you combine them into one equation? What can you do with this equation? How would you know if this method would work?y=3x-2y=x+4Leaner/Decision Maker: What do students do once they’ve found x? Do they go back and find y? The Tug of War gives us the experience of using equivalence to make sense of connected relationships, but when working with systems of equations we need to make sure we have the whole solution and not just one part of it. How do you want to approach this discussion as a class? What misconceptions do you see in your students work? Activity 3: You will now practice solving system of equations through both graphing and substitution.Note to Teacher: Here are some other problems you can do with your students.Questioner: When does substitution make sense? When doesn’t it make sense? Solve each system first by graphing and then algebraically.36861755588000a. y = 4x – 1 y=-12x+84257675-7683500b . y = 3x+2 2x+y=7454342514287500c. y=2x+1x-y=7both methods?orderbstcallyes that 2 and y = ese two equations? and several changed have been made t each453390032194500d. 3x+y=53x+y=8Journal Writing: What are similarities and differences between solving a system of equations graphically and by substitution? What are some skills you have to use in order to use both methods?Homework: Note to Teacher: This is exam item 2 from the August 2015 CC Algebra I Regents exam.Here is the original question. We made it into an open-ended problem thinking that much more can be learned from the problem in this formThis is exam item 2 from the August 2015 CC Algebra I Regents exam.Write as many true statements as you can about this relationship.6115055080000Rowan has $50 in a savings jar and is putting in $5 every week. Jonah has $10 in his own jar and is putting in $15 each week. Each of them plots his progress on a graph with time on the horizontal axis and amount of money in the jar on the vertical axis. Unit 3: LinearityLesson 14: Solving Systems of Equations by SubstitutionStudent VersionName_______________________Date________________________Opening Activity: Tug of WarFour oxen are as strong as five horses.60020202724150053898803143250047688502806700040811452806700032340552806700024117303060700016065502895600075946030607000 An elephant is as strong as one ox and two horses437515046037500386334048577500276415535941000 Who will win the tug-of-war pictured below? 585978042989500501269043878500418211044704000309943542164000242887545529500179387546355000114490539687500 Activity 2: Using the same two equations from yesterday’s Opening Activity, how could you combine them into one equation? What can you do with this equation? How would you know if this method would work?y=3x-2y=x+4Activity 3: You will now practice solving system of equations through both graphing and substitution.Solve each system first by graphing and then algebraically.36861755588000a. y = 4x – 1 y=-12x+84257675-7683500b . y = 3x+2 2x+y=7454342514287500c. y=2x+1x-y=7both methods?orderbstcallyes that 2 and y = ese two equations? and several changed have been made t each453390032194500d. 3x+y=53x+y=8Journal Writing: What are similarities and differences between solving a system of equations graphically and by substitution? What are some skills you have to use in order to use both methods?Homework: Note to Teacher: This is exam item 2 from the August 2015 CC Algebra I Regents exam.Here is the original question. We made it into an open-ended problem thinking that much more can be learned from the problem in this formThis is exam item 2 from the August 2015 CC Algebra I Regents exam.Write as many true statements as you can about this relationship.6057905334000Rowan has $50 in a savings jar and is putting in $5 every week. Jonah has $10 in his own jar and is putting in $15 each week. Each of them plots his progress on a graph with time on the horizontal axis and amount of money in the jar on the vertical axis. Unit 3: LinearityLesson 15: Solving Systems of Equations Using EliminationTeacher GuideDecision Maker: Throughout this lesson students will have the opportunity to take a go at a system of equations and to try to engage with trying to solve it. Through this you’ll get to learn about what sense your students made of their middle school experience. During this lesson you’ll need to decide when and how to transition in and out of discussion about the ideas present. Remember to keep the students thinking central and give them opportunities to address one another’s misunderstandings first before you enter into the discussion.Opening Activity:Learner: How do your students approach this system? What misunderstandings does their work expose?Look at the system of equations below. Try to solve it in any way you choosex-y=12x+y=6Activity 2:Provocateur: The work below contains a meaningful transition for students in their understanding of a basic operation. The use of addition here moves beyond real numbers and being used to manipulate an equation, to operating on equations! Together with all we’ve been talking about with solution, this is a new and amazing transition in our understanding of how addition and subtraction can be used. Be sure to draw the students’ attention to this in the discussion that follows Activity 2. Helping students grasp these types of transitions helps them begin to see the coherence and structure of mathematics.Now you are going to look at a different approach to solving this problem. Take a look. Do you understand what is happening? Can you explain the why this method might make sense? x – y = 12 x + y = 6 2x = 18 x = 9 x + y = 6 9 + y = 6 -9 = -9 y = -3 The solution to this system of equations is (9, -3).Questioner: How do we know this solution makes both of these equations true? What did we work with in Unit 2 that can help us make sense of what’s going on here? Why does making the coefficient on y 0 helpful? How is the new equation similar to other equations we’ve previously worked with? Activity 3: Take a look at the following system of equations. How might you solve this one?2x-y=6x+3y=10Questioner: Does what was done in Activity 2 work here? Why not? Is there something that could be done so that we could use the approach taken in Activity 2?Activity 4:Provocateur: The work below contains a transition for students in their understanding of multiplication. The use of multiplication here now includes operating on the equation as a whole. Admittedly, this transition isn’t as dramatic as the one with addition, but it’s worth making note of with your students. You can ask if 3(2x-y) = (6)3 is the same thing? If so, is this really all that different from what we’ve done when solving? This type of discussion though can help mathematics become more malleable and less rigid to students.Learner: How do your students approach this system? What misunderstandings does their work expose?Learner: What sense do the students make of this work? What misunderstandings do you see in your students’ sense making? How can these be incorporated in the class discussion at the end of this activity?Now you are going to look at a different approach to solving this problem. Take a look. Do you understand what is happening? Can you explain why this method might make sense? If you feel confused, what is confusing you? What questions would like to ask the class? 2x – y = 6 x + 3y = 10 3(2x – y = 6) x + 3y = 10 6x – 3y = 18 x + 3y = 10 7x = 28 x = 4 x + 3y =10 4 + 3y = 10 3y = 6 y = 2 The solution to this system of equations is (4, 2).Questioner: What did this student do initially? What did that enable us to do? Activity 5:Learner: Do your students see the connection from Activity 4 to this activity? Questioner: How can we use what was discussed in Activity 4 to help us sort out this situation?One of the systems of equations has the same solution as the system below. Which one is it? How do you know?2x+2y=163x-y=4a) 2x+2y=166x-2y=4b) 2x+2y=166x-2y=8 c) x+y=16 3x-y=4d) 6x+6y=486x+2y=8Activity 6:Decision Maker: What do you want your students to get out of this activity? Can grouping the students differently help?Now you are going to practice with solving systems of equations. You can choose any method you want.x-y=6x+y=103x-2y=15x+y=10y=3x-12 x+y=10Unit 3: LinearityLesson 15: Solving Systems of Equations Using EliminationStudent VersionName_______________________Date________________________Opening Activity:Look at the system of equations below. Try to solve it in any way you choosex-y=12x+y=6Activity 2:Now you are going to look at a different approach to solving this problem. Take a look. Do you understand what is happening? Can you explain the why this method might make sense? x – y = 12 x + y = 6 2x = 18 x = 9 x + y = 6 9 + y = 6 -9 = -9 y = -3 The solution to this system of equations is (9, -3).Activity 3: Take a look at the following system of equations. How might you solve this one?2x-y=6x+3y=10Activity 4:Now you are going to look at a different approach to solving this problem. Take a look. Do you understand what is happening? Can you explain the why this method might make sense? If you feel confused, what is confusing you? What questions would like to ask to the class? 2x – y = 6 x + 3y = 10 3(2x – y = 6) x + 3y = 10 6x – 3y = 18 x + 3y = 10 7x = 28 x = 4 x + 3y =10 4 + 3y = 10 3y = 6 y = 2 The solution to this system of equations is (4, 2).Activity 5: One of the systems equations has the same solution as the system below. Which one is it? How do you know?2x+2y=163x-y=4a) 2x+2y=166x-2y=4b) 2x+2y=166x-2y=8 c) x+y=16 3x-y=4d) 6x+6y=486x+2y=8Activity 6: Now you are going to practice with solving systems of equations. You can choose any method you want.x-y=6x+y=103x-2y=15x+y=10y=3x-12 x+y=10Unit 3: LinearityLessons 16 & 17: Solving Systems of Equations Contextual ProblemsTeacher GuideCoach: The next two lessons are about how students work with problems and use a systems of equations approach to solving them. You can use the KNN charts to help students organize their thinking. You will be given five problems that students can work on over the next two days. They can be solved in multiple ways. You should work with each of them and think about how students might try to solve them.Artist: How do you want to facilitate the student experiences for the next two lessons? Do you want students to work on one at a time then you can have a class discussion led by the student ideas?Do you want students to work on all of them and then focus on one of them (based on your selection) that they would present to the rest of the class? Do you want students to be able to show at least two methods to solve their selected problem?Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext StepsProblem 1: Find the missing numbersFind two numbers such that the sum of the first and three times the second is 5 and the sum of second and two times the first is 8. Problem 2: Tweedledum and TweedledeeIn Lewis Carroll’s Through the Looking Glass, Tweedledum says, “The sum of your weight and twice mine is 361 pounds.” Tweedledee replies, “The sum of your weight and twice mine is 362 pounds.” Find both of their weights.Further Question to be ready to discuss:Can you solve this only using Tweedledum’s statement? Why? Learner: The key idea here is you want the students to think about the fact that each statement alone has infinite amount of solutions but when you bring them together you only have one solution.)Problem 3: Lulu’s RiddleCoach: This problem can be solved in different ways: guess and check, using one variable, or two variables. Again let your students play with them in any way they choose. Then you can examine the different methods, asking students to think about: Which approach do you prefer? Why?Learner: Students have historically struggled with this type of problem. The issue that causes problems is representing the value of the money in the equation. Students also struggle with the two equations; they each represent a different idea: x + y = 20 represents quantity of coins, while .10x + .25y = 4.10 represents value of the coins. This can be confusing yet we can work with them together because the x and y represent the same thing in both equations.Lulu tells her little brother, Jack, that she is holding 20 coins all of which are dimes and quarters. They have a value of $4.10. She says she will give him the coins if he can tell her how many of each she is holding. How can Jack go about figuring out Lulu’s riddle?Problem 4: Twin Brothers’ RivalryCoach: The following problem is written in the form it would probably look like on a state exam or a PARCC/Smarter Balance exam. It is not a pure problem since it is telling students what to do and thus is less open ended. But try it with them. They are asked to solve it graphically which will be interesting to see what students do with that.Andy’s Cab Service charges a $6 fee plus $0.50 per mile. His twin brother Randy starts a rival business where he charges $0.80 per mile, but does not charge a fee. Write a cost equation for each cab service in terms of the number of miles.Graph both cost equations. For what trip distances should a customer use Andy’s Cab Service? For what trip distances should a customer use Randy’s Cab Service? Justify your answer algebraically and show the location of the solution on the graph.Problem 5: Pam’s part time jobs.Pam has two part time jobs. At one job, she works as a cashier and makes $8 per hour. At the second job, she works as a tutor and makes $12 per hour. One week she worked 30 hours and made $268. How many hours did she spend at each job?left280606500Problem 6: Albert’s Dilemma1035058255000Unit 3: LinearityLesson 16 & 17: Solving Systems of Equations Contextual ProblemsStudent VersionProblem 1: Find the missing numbersFind two numbers such that the sum of the first and three times the second is 5 and the sum of second and two times the first is 8. Problem 2: Tweedledum and TweedledeeIn Lewis Carroll’s Through the Looking Glass, Tweedledum says, “The sum of your weight and twice mine is 361 pounds.” Tweedledee replies, “The sum of your weight and twice mine is 362 pounds.” Find both of their weights.Problem 3: Lulu’s RiddleLulu tells her little brother, Jack, that she is holding 20 coins all of which are dimes and quarters. They have a value of $4.10. She says she will give him the coins if he can tell her how many of each she is holding. How can Jack go about figuring out Lulu’s riddle?Problem 4: Twin Brothers’ RivalryAndy’s Cab Service charges a $6 fee plus $0.50 per mile. His twin brother Randy starts a rival business where he charges $0.80 per mile, but does not charge a fee. Write a cost equation for each cab service in terms of the number of miles.Graph both cost equations. For what trip distances should a customer use Andy’s Cab Service? For what trip distances should a customer use Randy’s Cab Service? Justify your answer algebraically and show the location of the solution on the graph.Problem 5: Pam’s part time jobs. Pam has two part time jobs. At one job, she works as a cashier and makes $8 per hour. At the second job, she works as a tutor and makes $12 per hour. One week she worked 30 hours and made $268. How many hours did she spend at each job?Unit 3: LinearityLesson 18: The Hamburger ProblemTeacher GuideDecision Maker: This is a rich problem where students can learn that there is much they can see with limited information. How do you want to facilitate the student activity? How might you group the students? Do you want to start with the KNN chart? It might help to get everyone started. When do you want to bring the whole class together?It will be important to observe what groups are doing so you can get a better sense of what your students understand about systems of equations and use that information to make better decisions about instruction.Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext Steps2196465106680000The Hamburger Problem Sue Flay opened a MacDonald’s on White Plains Road and Cassa Role opened a Burger King across the street. Both had to borrow money to open their fast food franchises. After 500 customers, Sue was still $4000 in debt. By the time she had served 3000 customers, she was ahead $1000.After 2000 customers, Cassa Role still owed $6000 to the bank. However, after 4500 customers, she was ahead by $1500.Which restaurant would you rather own? (Hint: At what points do you want to own them?)Support your answer with mathematical evidenceInclude in your solution:Amount each had to borrow to open the storeAmount of customers they needed to break even Unit 3: Linearity Student VersionLesson 1854483002730500:The Hamburger Problem Sue Flay opened a MacDonald’s on White Plains Road and Cassa Role opened a Burger King across the street. Both had to borrow money to open their fast food franchises. After 500 customers, Sue was still $4000 in debt. By the time she had served 3000 customers, she was ahead $1000.After 2000 customers, Cassa Role still owed $6000 to the bank. However, after 4500 customers, she was ahead by $1500.Which restaurant would you rather own? (Hint: At what points do you want to own them?)Support your answer with mathematical evidenceInclude in your solution:Amount each had to borrow to open the storeAmount of customers they needed to break evenUnit 3: LinearityLesson 19: The RaceTeacher Guide441007510096500The RaceDecision Maker: Do you want to make this an on-demand task? Do you want students working individually or in pairs?Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?KnowNeed to KnowNext StepsSitting in his math class, Malik is trying to figure out how he is going to get an A for the year, especially since he has missed handing in so much of the homework. Then, he has a brilliant idea.Malik’s teacher, Mr./Ms. _________________________ is always bragging about what a(write your teacher’s name here)great runner s/he was in college and how many trophies s/he won. So Malik says to his teacher, “I have a cousin who is a pretty good runner. Would you like to race him? He’s a little bit younger than you so I bet he would even give you a head start.” Malik dares his teacher and says, “If my cousin beats you, you give me an A in math for the year. If you win, I’ll make up all my homework and do any extra work that you give me.” Malik’s teacher thinks for a minute, “After I win, I’ll be able to get Malik to catch up on his work.” Then s/he says, “Okay. I’ll do it. Who’s your cousin?”Malik: “Usain Bolt.”Teacher: “Usain Bolt! You mean the Jamaican guy who made world records for the 100 and 200 meter races in the 2008 and 2012 Olympics?”Malik: “That’s rrriiigghhttt. But you can’t back out now.” Teacher: “Okay, I’ll do it. But I better get a good head start.”Malik: “Okay. I’ll lay out the race for you and I’ll make sure you have a good head start.”The Day of the RaceWhen the starting buzzer sounds, Malik’s teacher springs from the starting line and tears down the course. Usain takes off some time later.NOTE: ALL TIMES ARE GIVEN SINCE THE STARTING BUZZER SOUNDED.-8699514922500The Teacher:15 seconds after the buzzer, the teacher still has 135 meters to go to the finish line. 32 seconds after he left the starting line, the teacher has only 50 meters more to run.-8699516637000Usain Bolt (Usain starts running at some point after the teacher starts)22 seconds after the buzzer, Usain is 189 meters from the finish line. 36 seconds after the buzzer, he is 42 meters from the finish line.BOTH PEOPLE ARE RUNNING AT A CONSTANT SPEED. Your task is to determine the outcome of the race. You need to explain the process you use and show all the mathematics. After you’ve finished, try to support your thinking by using another method.Show all your work:Unit 3: Linearity Student Version46767758191500Lesson 19:The RaceSitting in his math class, Malik is trying to figure out how he is going to get an A for the year, especially since he has missed handing in so much of the homework. Then, he has a brilliant idea.Malik’s teacher, Mr./Ms. _________________________ is always bragging about what a(write your teacher’s name here)great runner s/he was in college and how many trophies s/he won. So Malik says to his teacher, “I have a cousin who is a pretty good runner. Would you like to race him? He’s a little bit younger than you so I bet he would even give you a head start.” Malik dares his teacher and says, “If my cousin beats you, you give me an A in math for the year. If you win, I’ll make up all my homework and do any extra work that you give me.” Malik’s teacher thinks for a minute, “After I win, I’ll be able to get Malik to catch up on his work.” Then s/he says, “Okay. I’ll do it. Who’s your cousin?”Malik: “Usain Bolt.”Teacher: “Usain Bolt! You mean the Jamaican guy who made world records for the 100 and 200 meter races in the 2008 and 2012 Olympics?”Malik: “That’s rrriiigghhttt. But you can’t back out now.” Teacher: “Okay, I’ll do it. But I better get a good head start.”Malik: “Okay. I’ll lay out the race for you and I’ll make sure you have a good head start.”The Day of the RaceWhen the starting buzzer sounds, Malik’s teacher springs from the starting line and tears down the course. Usain takes off some time later.NOTE: ALL TIMES ARE GIVEN SINCE THE STARTING BUZZER SOUNDED.-8699514922500The Teacher:15 seconds after the buzzer, the teacher still has 135 meters to go to the finish line. 32 seconds after he left the starting line, the teacher has only 50 meters more to run.-8699516637000Usain Bolt (Usain starts running at some point after the teacher starts)22 seconds after the buzzer, Usain is 189 meters from the finish line. 36 seconds after the buzzer, he is 42 meters from the finish line.BOTH PEOPLE ARE RUNNING AT A CONSTANT SPEED. Your task is to determine the outcome of the race. You need to explain the process you use and show all the mathematics. After you’ve finished, try to support your thinking by using another method.Show all your work:Unit 3: LinearityLesson 20: Graphing InequalitiesTeacher GuideOpening Activity:Learner: What approach do students turn to in working through this problem? Do they try to graph? Do they substitute?-3619513716000Circle each ordered pair that is a solution to the equation y > 4x - 10. (3,2)(2,3)(-1, -14)(0,0)(1, -6)(5, 10)(0, -10)(3, 4)(6, 0)(4, -1)(0, 8)(-2, 1)(5, 2)(3, 6)(-6, 12)(-8, 14)(-9,0)Activity 2: Plot all the solutions on the coordinate plane. Then graph the equation y > 4x - 10. What do you observe about the relationship between the solution points and the graph of the line?Decision Maker: This is your opportunity to have a discussion about solution sets to an inequality and how we graph them. It is also an opportunity to relate the inequality symbol to the idea of “above” and “below”. Students can also test points and see which do and don’t satisfy the inequality..Activity 3: Practice with graphing inequalitiesDecision Maker: What inequalities do you want to use? Here are four inequalities you could use but you should make the final call on what you think will be best for your students to experience.y < -3x + 2y > 2x – 3y ≤ 5x – 2y ≥ -4x + 1Unit 3: LinearityLesson 20: Graphing InequalitiesStudent VersionName_______________________Date________________________Opening Activity:-3619513716000Circle each ordered pair that is a solution to the equation y > 4x - 10. (3,2)(2,3)(-1, -14)(0,0)(1, -6)(5, 10)(0, -10)(3, 4)(6, 0)(4, -1)(0, 8)(-2, 1)(5, 2)(3, 6)(-6, 12)(-8, 14)(-9,0)Activity 2: Plot all the solutions on the coordinate plane. Then graph the equation y > 4x - 10. What do you observe about the relationship between the solution points and the graph of the line? Activity 3: Practice with graphing inequalitiesy < -3x + 2y > 2x – 3y ≤ 5x – 2y ≥ -4x + 1Unit 3: LinearityLesson 21: Graphing Systems of InequalitiesTeacher GuideOpening Activity:Decision Maker: How do you want to facilitate the Opening Activity? There is a lot in this activity and you’ll need to decide when you want to have full class discussions.Provocateur: This Opening Activity is provocative, because we are asking students to graph an equation and an inequality. How does this effect solution?Consider the following compound sentence: x + y > 10 AND y = 2x + 1.Decision Maker: As you prepare for this lesson make sure you think about the graph because it will impact the decisions you make and the questions you ask during the lesson.302196520637500952546672500A list of ordered pairs, (x, y), are provided below. Circle the ones that are solutions to x + y > 10. Also, underline those that are solutions to y = 2x + 1.3,77, 3-1,140,112,255,110, 121,812,0-1,-1Which ordered pair(s) from above can make the compound statement true x + y > 10 AND y = 2x + 1.Create three more points that make the compound statement true.Decision Maker: This could be a good time to have a discussion. How do you know your points work? What’s your basis?How many possible answers are there? How do you know?Provocateur: You are trying to get at the idea of what solutions look like in systems of inequalities. Sketch the solution set to the inequality x + y > 10 and the solution set to y = 2x + 1 on the same set of coordinate axes. Highlight the points that lie in BOTH solution sets.Describe the solution set to x + y > 10 AND y = 2x + 1.On your graph plot the points that were circled AND underlined in the first part of this activity. Where did these plotted points end up? Does this make sense?Which gives a more clear idea of the solution set – the graph or the verbal description? How does the solution set change if the equality (y=2x+1) is changed into y > 2x+1?Activity 2:Provocateur: How do students move from the solution from the original problem to the solution now that has two inequalities?Now suppose the system of equations from Opening Activity was instead a system of inequalities:x+y>102x+1<y How will the change affect your graph?View in Desmos: 3:Decision Maker: Instruct students to graph and shade the solution set to each inequality in two different colored pencils. Give them a few minutes to complete this individually. Then, discuss the solution to the system as a class.Graph the solution set to the system of inequalities. 2x – y < 3 and 4x + 3y ≥ 0Where does the solution to the system of inequalities lie? What is true about all of the points in this region? Verify this by testing a couple of points from the shaded region and a couple of points that are not in the shaded region to confirm this idea to students.Closing Activity:Learner: Have students work on the following and be prepared to talk about their thinking. (The second one is most interesting. What would students do with this?Graph the solution set to each system of inequalities.x-y>5x>-1y≤x+4y≤4-xy≥0Unit 3: LinearityLesson 21: Graphing Systems of InequalitiesStudent VersionName_______________________Date________________________Opening Activity:Consider the following compound sentence: x + y > 10 AND y = 2x + 1.301815521336000952546672500A list of ordered pairs, (x, y), are provided below. Circle the ones that are solutions to x + y > 10. Also, underline those that are solutions to y = 2x + 1.3,77, 3-1,140,112,255,110, 121,812,0-1,-1Which ordered pair(s) from above can make the compound statement true x + y > 10 AND y = 2x + 1.Create three more points that make the compound statement true.How many possible answers are there? How do you know?Sketch the solution set to the inequality x + y > 10 and the solution set to y = 2x + 1 on the same set of coordinate axes. Highlight the points that lie in BOTH solution sets.Describe the solution set to x + y > 10 AND y = 2x + 1.On your graph plot the points that were circled AND underlined in the first part of this activity. Where did these plotted points end up? Does this make sense?Which gives a more clear idea of the solution set – the graph or the verbal description? Activity 2:Now suppose the system of equations from Opening Activity was instead a system of inequalities:x+y>102x+1<y How will the change affect your graph?Activity 3:Graph the solution set to the system of inequalities. 2x – y < 3 and 4x + 3y ≥ 0Where does the solution to the system of inequalities lie? What is true about all of the points in this region? Verify this by testing a couple of points from the shaded region and a couple of points that are not in the shaded region to confirm this idea.Closing Activity: Graph the solution set to each system of inequalities.x-y>5x>-1y≤x+4y≤4-xy≥0Unit 3: LinearityLesson 22: Solving System of Inequality Contextual ProblemsTeacher GuideCoach: Each of these problems will require the students to construct and graph an inequality. In conversations with students ask how other problems in this unit can help us make sense of these. Can the problem be simplified to get a better sense of the problem?Decision Maker: Will you provide your students with a KNN chart? Or will you make that a suggestion as you coach/question your students?5205730218884500KnowNeed to KnowNext StepsProblem 1: Drama for the Drama Club? A high school drama club is putting on their annual theater production. There is a maximum of 800 tickets for the show. The costs of the tickets are $6 before the day of the show and $9 on the day of the show. To meet the expenses of the show, the club must sell at least $5,000 worth of tickets. The Drama Club expects to sell most of their tickets before the day of the show.Based on your graph, can you give 2 solutions and 2 non-solutions? Explain how you know What’s the minimum amount of tickets they have to sell at the door to break even?Problem 2: Cost of Shelter502920041529000An animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. The shelter can spend at most $89.50 caring for cats and dogs on Wednesday. At most the shelter can handle 35 cats and dogs in total.If the shelter were to get only dogs, what’s the maximum number of dogs they could shelter?If the shelter were to get only cats, what’s the maximum number of cats they could shelter?Provocateur: How do students answer this question? According to the price the shelter can take 38 cats, but according to space it can only handle 35. How do the students account for the context?Problem 3: PJs & T-shirtsA clothing manufacturer has 1000 yd. of cotton to make shirts and pajamas. A shirt requires 1 yd. of fabric and a pair of pajamas requires 2 yd. of fabric. It takes 2 hr. to make a shirt and 3 hr. to make the pajamas, and there are 1600 hr. available to make the clothing. How many shirts and pajamas would you recommend the manufacturer make in order to make the best use of materials and time?Unit 3: LinearityLesson 22: Solving System of Inequality Contextual ProblemsStudent Version641921511239500Problem 1: Drama for the Drama Club? A high school drama club is putting on their annual theater production. There is a maximum of 800 tickets for the show. The costs of the tickets are $6 before the day of the show and $9 on the day of the show. To meet the expenses of the show, the club must sell at least $5,000 worth of tickets. The Drama Club expects to sell most of their tickets before the day of the show.Based on your graph, can you give 2 solutions and 2 non-solutions? Explain how you know What’s the minimum amount of tickets they have to sell at the door to break even?Problem 2: Cost of ShelterAn animal shelter spends $2.35 per day to care for each cat and $5.50 per day to care for each dog. The shelter can spend at most $89.50 caring for cats and dogs on Wednesday. At most the shelter can handle 35 cats and dogs in total.If the shelter were to get only dogs, what’s the maximum number of dogs they could shelter?If the shelter were to get only cats, what’s the maximum number of cats they could shelter?center191833500Problem 3: PJs & T-shirtsA clothing manufacturer has 1000 yd. of cotton to make shirts and pajamas. A shirt requires 1 yd. of fabric and a pair of pajamas requires 2 yd. of fabric. It takes 2 hr. to make a shirt and 3 hr. to make the pajamas, and there are 1600 hr. available to make the clothing. How many shirts and pajamas would you recommend the manufacturer make in order to make the best use of materials and time?Final Project/Reflection:Learner: What do you want to learn about your students from this end of unit? Here are some ideas to consider.Part I. Create a story about this distance versus time graph. Consider if this were a speed vs. time graph. How would your story change? Part 2: Create a system of 3 inequalities in which (3, 4) is a solution.Regents Questions.Artist: How will you structure exposure to Regents exam items? Mitsouka Jean Claude, teacher at Victory Collegiate HS, and her math coach created a Tic Tac Toe structure* to Regents practice. It gives students choice and encourages students to try different problems.* 2015, #5 August 2015, #2 August 2015, #10left4127500left35433000August 2015, #6June 2015, #6June 2015, #5118110698500August 2015, #26left-17589500June 2015, #2June 2015, #2left3111500January 2015, #9left13271500August 2015, #28left14351000January 2015, #116350381000June 2015, #356350698500left12573000January 2015, #33left1841500August 2014, #7left1143000January 2015, #346350000left1714500August 2014, #86350000left26543000August 2014, #13left33020000August 2014, #37left952500Determine and state one combination of hours that will allow Edith to earn at least $80 per week while working no more than 15 hours.August 2014, #257152921000August 2014, #20June 2014, #46350000June 2014, #14left4445000left50101500June 2014, #18June 2014, #29left12827000Is the point (3,2) a solution to the equation? Explain your answer based on the graph drawn. ................
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