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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 2: Equality, Inequality & Problem SolvingAligned to the Common Core Standardstc "Unit 2: Equality, Inequality & Problem Solving" \l 1Unit 2: Equality, Inequality & Problem SolvingThe large purpose of this unit is to develop a deeper and nuanced understanding of equality. Equality is what makes the properties the properties at the beginning of the unit and is the central idea that allows for flexibility in solving equations at the end of the unit. Throughout the curriculum equality will continue to be a central idea to the development of mathematical thinking. tc "Unit 2 Outline" \l 2Essential Questions: Why is it important to think about the underlying structures of mathematics?What is the meaning of “equality” and why is this understanding essential to procedural understanding? ?Interim Assessments/Performance TasksMystery of the Two Coupons Create Three EquationsAm I a Flexible Thinker? 4 VignettesMaria’s Dilemma?Final Assessment:Great Adventures and other problems?What will students understand and be able to do at the end of the unit?Students will?develop deeper understanding of mathematical properties and their meaning within algebra. Students will understand the meaning of equations and will have greater flexibility in working with them.Students will understand inequalities with greater depth and be able to work with them using multiple representations.Students will begin to understand how to solve problems using an algebraic approach Students will be able to solve different types of problems.?What enduring understanding will students have?The properties/laws/rules of mathematics can be understood and are part of the essential structure of mathematics. An algebraic approach to solving problems can be an efficient and powerful way to solve problems.Ideas of inequalities are based on our understanding of equations but new ideas need to be thought about (e.g. meaning of solution, graphing, and approaches to solving) when thinking about them.Standards Addressed:A-CED.1 Create equations and inequalities in one variable and use them to solve problems.A-CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.A-REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.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-CED.A.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.?For example, rearrange Ohm's law V = IR to highlight resistance R.Lesson BookmarksLesson 1: Does order matter?Lesson 2: Are both methods equivalent?Lesson 3: Two Coupons MysteryLesson 4: Four Laws of EqualityLesson 5: What is the meaning of equality?Lesson 6: Solving Equations, Maintaining Equivalence & JustificationLesson 7: Practice with equations, work with errorsLesson 8: Am I a flexible Thinker: 4 VignettesLesson 9: Literal Equations and Rearranging FormulasLesson 10: Review of Inequalities LessonLesson 11: AND & OR Lesson 12: How does the domain change our thinking and affect the solution sets?Lesson 13: Can you discover how to solve inequalities?Lessons 14: Solving Compound InequalitiesLessons 15 – 19: Algebraic Problem SolvingLesson 20: Inequality ProblemsLesson 21: Final Performance Tasks/ProjectAppendix A: Two Connected Inequality LessonsUnit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Note to teacher: Here* is a link to an optional (mini) number sense unit. It examines fractions, rational/irrational numbers and the uniqueness of finite and infinity with the lens of students becoming better pattern hunters. * our notion of equality at the end of this unit is focused on solving equations, here at the outset we are using the notion of equality in talking about the different properties. So when one works with properties equality is the underlying idea. One aspect of equality to emphasize within the properties is that though representations look different they have the same value. For example: 5 = 2 + 3 = (2 + 1) + 2 = 4 + 1 = 1 + (1 + 3) and so on. Lesson 1: Does order matter?Teacher GuideNote to Teacher: In today’s activities you will be helping students develop a deeper understanding of the commutative and associative properties. These properties allow tremendous freedom in computations. Based on their understanding of the variable students should be able to generalize arithmetic ideas. The following activities should be performed in groups. You should bring the class together after each activity to lead a discussion about their findings and understanding.Opening ActivityCoach: In the opening activity we’re trying to focus the students’ attention to behavior, because the properties of commutation and associativity can be thought of as descriptors of the behaviors. Select the question that is most appealing to you. Write your opinion in response to the question.Do you think boys behave differently around other boys than they do girls? If so describe the difference in that behavior and give it a name.Do you think girls behave differently around other girls than they do boys? If so describe that behavior and give it a name.Do you think teenagers behave differently when they are with their friends than when they are with their families? If so describe that behavior and give it a name.Do you think high school students behave differently at school than they do outside school? If so describe that behavior and give it a name.Be prepared to talk about the question you chose.Reflector: Have students share out their thinking on these questions: How did you describe the change in behavior? How did you come up with the name? This sequence of observing, describing and naming behavior is basically what the students will be doing in the rest of the lesson. They will observe how the operations behave in different situations, describe that behavior, and give a name for that behavior.Coach: Before transitioning into the next activity you could point out that what they did in this opening activity, generalizing a human behavior and naming it, is a focus in psychology. Also, if this is something that interests them they could consider it as a major in college.Activity 2:Coach: As human beings we all behave differently in different situations. How do we behave in the three situations below? In two of them we will always do one before the other. In one of them it could differ each day.Look at the three events given below. You need to determine if the order in which you do these activities will affect the overall result. In other words, can you do either a first or b first and still get the same result?a. Scrape the dinner dishesb. Wash the dinner dishesWash your faceBrush your teethDrain the used oil from the car’s engineAdd four quarts of new oil1) Which of these events would work best as a model for addition of two numbers? Explain your answer.2) If a and b represent any two numbers is it always true that a + b = b + a? Explain.Decision Maker: You need to give this property its name. Consider asking the students, “Why would you need to name this characteristic of addition?” Communication is an important and often overlooked aspect of mathematics. Through the discussion help the students understand that mathematical vocabulary is used to generalize whole mathematical concepts for increased efficiency in our communication. This way we can use the vocabulary instead of non-specific references like, “that thing” or “that behavior of addition”. Additionally, giving things a name acts in a similar fashion to the variable. It allows us to speak generally of every instance of that specific concept. Talking about these ideas will help answer the perpetual question of “Why?” and hopefully help expand their understanding of the variable. Show them the Commutative Law for Addition: a + b = b + a.Activity 3: Fill in the following table:Decision Maker: Feel free to change the values in the table as you see fit. However, with the values in the table it would be good for students to have the opportunity to realize when commutation holds true for subtraction and division as well as what’s special about those cases (for example, the commutative property holds for subtraction when a and b are the same value). This will also set you up to ask some additional questions to students as you are entering into their investigation.aba+bb+aa-bb-aa?bb?aa÷bb÷a12223343441.536-35-5510Look at the data you collected in your table. Write your observations.Coach: A way to help students focus their observations is to ask them, “What is the focus of filling out this table?” Some students will have astute answers to this question, while other may need some more help. We want to help students to grow in their ability to make observations that are meaningful to the task at hand. This is where the focusing on the idea of behavior in the opening activity comes in handy, because here we really want to make observations about how the operations are behaving when we switch the numbers being evaluated.What is your conjecture about the different operations when you switch the order of the numbers? Will your conjecture work with fractions? Decimals? To be able to prove a law doesn’t work in mathematics, you just need one example. Give an example for each of the operations for which this law doesn’t work.Make a general statement for when the commutative property (changing the order of the numbers) holds for a given operation, e.g. addition, subtraction, multiplication, division?Coach: To help students understand why we name these properties as we do, ask students what it means to commute, associate, and distribute?The commutative property says that you can change the order of the numbers for a given operation and still get the same result. Activity 4:Now let’s look at another property.1) Are these true????4 + (5 + 6) = (4 + 5) + 61 + (8 + 3) = (1 + 8) + 39 + (7 + 2) = (9 + 7) + 2Note to Teacher: The above is formatted using a borderless table.2) If you replaced the numbers with different numbers would it still be true? Justify your answer.3) Can you find an example that won’t work? This called the associative property. It says, we can group numbers differently and still get the same result. Describe how this definition of the property is related to the three examples you worked with.Try this property with the other three operations.Subtraction: Do you think the associative property is true for subtraction? Why or why not?Test it out:?4 – (5 – 6) = (4 – 5) – 6Multiplication: Do you think the associative property is true for multiplication? Why or why not?Test it out:?4 ? (5 ? 6) = (4 ? 5) ? 6Division: Do you think the associative property is true for division? Why or why not?Test it out:?4 ÷ (5 ÷ 6) = (4 ÷ 5) ÷ 6Observation: What connections do you see among the operations that the associative property holds true for and among those that it doesn’t? How would you explain this connection?Provocateur: Students could make any number of observations about the connection between addition and multiplication and the associative property holding true for them, but not for division and subtraction. However, what’s most interesting to consider here is the connection between the commutative property and the associative property. Could the associative hold true if the commutative property didn’t hold true for a given operation? It would be a wonderful question to consider, but not resolve with students.Conclusion: What general statement can you make now about the associative property and the four operations?Reflector: Have the students share out their thoughts on the connections as well as their general statements. Memorialize some of their general statements to keep up in the classroom.Note to teacher: We want students to gain an understanding that the properties are a natural part of what we do in our minds when we do mathematics (arithmetic, algebraic, or otherwise). The essence of the properties allow us make decisions in our heads.Exit Ticket/Closing Activity: Mr. Gauss was given the following calculation: 6.1 + (8 + 3.9) and was asked to do it in his head.Here is his mental work:6.1 + (8 + 3.9) = 6.1 + (3.9 + 8)= (6.1 + 3.9) + 8= 10 + 8= 18Explain what he did in each step and what justifications he would give to making those decisions.Decision Maker: Here is a combinations problem that relies on the idea of commutativity: 1 + 1 + 23 = 1 + 23 + 1 = 23+ 1 + 1. It is surprising to discover that there are only 16 different combinations. This is yet another opportunity for students to practice strategizing an approach, organizing a strategy and looking for patterns that would be facilitated by the use of a table. 4824152-12106700Homework: Penny’s Dimes Penny has 25 dimes. She likes to arrange them into three piles, putting an odd number of dimes into each pile. In how many different ways could she do this?[Hint: Perhaps a KNN chart help you think about this problem.]What is the connection between this problem and today’s lesson?Unit 2: Equality, Inequality & Problem SolvingLesson 1: Does order matter?Student VersionName_______________________Date________________________Opening ActivitySelect the question that is most appealing to you. Write your opinion in response to the question.Do you think boys behave differently around other boys than they do girls? If so describe the difference in that behavior and give it a name.Do you think girls behave differently around other girls than they do boys? If so describe that behavior and give it a name.Do you think teenagers behave differently when they are with their friends than when they are with their families? If so describe that behavior and give it a name.Do you think high school students behave differently at school than they do outside school? If so describe that behavior and give it a name.Be prepared to talk about the question you chose.Activity 2:Look at the three events given below. You need to determine if the order in which you do these activities will affect the overall result. In other words, can you do either a first or b first and still get the same result?a. Scrape the dinner dishesb. Wash the dinner dishesWash your faceBrush your teethDrain the used oil from the car’s engineAdd four quarts of new oil1) Which of these events would work best as a model for addition of two numbers? Explain your answer.2) If a and b represent any two numbers is it always true that a + b = b + a? Explain.Activity 3: Fill in the following table:aba+bb+aa-bb-aa?bb?aa÷bb÷a12223343441.536-35-5510Look at the data you collected in your table. Write your observations.What is your conjecture about the different operations when you switch the order of the numbers? Will your conjecture work with fractions? Decimals? To be able to prove a law doesn’t work in mathematics, you just need one example.Give an example for each of the operations for which this law doesn’t work.Make a general statement for when the commutative property (changing the order of the numbers) holds for a given operation, e.g. addition, subtraction, multiplication, division?The commutative property says that you can change the order of the numbers for a given operation and still get the same result. Activity 4:Now let’s look at another property.1) Are these true????4 + (5 + 6) = (4 + 5) + 61 + (8 + 3) = (1 + 8) + 39 + (7 + 2) = (9 + 7) + 22) If you replaced the numbers with different numbers would it still be true? Justify your answer.3) Can you find an example that won’t work? This called the associative property. It says, we can group numbers differently and still get the same result.Describe how this definition of the property is related to the three examples you worked with.Try the associative property with the other three operations.Subtraction: Do you think the associative property is true for subtraction? Why or why not?Test it out:?4 – (5 – 6) = (4 – 5) – 6Multiplication: Do you think the associative property is true for multiplication? Why or why not?Test it out:?4 ? (5 ? 6) = (4 ? 5) ? 6Division: Do you think the associative property is true for division? Why or why not?Test it out:?4 ÷ (5 ÷ 6) = (4 ÷ 5) ÷ 6Observation: What connections do you see among the operations that the associative property holds true for and among those that it doesn’t? How would you explain this connection?Conclusion: What general statement can you make now about the associative property and the four operations?Exit Ticket/Closing Activity: Mr. Gauss was given the following calculation: 6.1 + (8 + 3.9) and was asked to do it in his head.Here is his mental work:6.1 + (8 + 3.9) = 6.1 (3.9 + 8)= (6.1 + 3.9) + 8= 10 + 8= 18Explain what he did in each step and what justifications he would give to making those decisions.4878686-12014200Homework: Penny’s Dimes Penny has 25 dimes. She likes to arrange them into three piles, putting an odd number of dimes into each pile. In how many different ways could she do this?[Hint: Perhaps a KNN chart help you think about this problem.]What is the connection between this problem and today’s lesson?Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 2: Are both methods equivalent?Teacher GuideOpening Activity19050128270You and your friend Shawn are selling boxes of candy on the bus/subway. You begin your speeches by saying, “I’m not going to lie to you. I’m not selling candy for my team, my church or any other organization. I’m selling this for me, to keep me off the street.” At the end of the week, your plan is to share the profits equally. For each box of candy sold, you and Shawn make $7. You sold 18 boxes and Shawn sold 23 boxes.Shawn’s method:Shawn chose to calculate the amount you and he will earn by adding the number of boxes sold and multiplying 7.Your method:You figured out your own earnings before you figured out Shawn’s earnings. Then you added your earnings and Shawn’s earnings pare your results with Shawn’s. What do you notice?Which method makes more sense to you? Why does it make more sense?1) Let m stand for the number of boxes you sold, s stand for the number Shawn sold and let c stand for the amount you earn on each box.Write an algebraic expression showing the method that Shawn used.Then write an algebraic expression showing the method that you used.Write an equation that includes the information from both of your methods.Coach: You’re looking for students to write: Shawn: c(m + s)Your method: cm + csc(m + s) = cm + csCoach: As you know, this third statement is the general equation for the distributive property of multiplication over addition. How do you ensure that students understand this important idea? Often, working with specific arithmetic values can help students to begin to make sense of the generalized form of the situation.2) Experiment with some other numbers to see if your equation holds true.Let:m = 4, s = + 6 , and c = 9Let:m = 3, s = -2, and c = -4Do you think this will hold true for any kind of numbers we use? What makes you think this?3) Do you think this property is true for multiplication over subtraction? Try several examples and report your conclusion.Activity 2:Note to Teacher: In this activity we are developing the identity property. It is another idea that students have seen before but one that is important in algebraic understanding and its relationship to the inverse property.Decision Maker: A differentiated version of Activities 2, 3 and 4 for ELLs and struggling students can be found here. ) Pick a number, any number. What would you add to this number so that the result does not change?Try this with a couple of other numbers.2a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How do you know?2b) Write a general equation. Use x represent to the numbers you and your classmates chose. Decision Maker: Let the students know that this is called the additive identity property and zero is the additive identity element.3) Pick a number, any number. What would you multiply this number by such that the result does not change?Try this with a couple of other numbers.4a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How do you?4b) Write a general equation. Use x represent to the numbers you and your classmates chose.Decision Maker: Let the students know that this is called the multiplicative identity property and one is the multiplicative identity element.Activity 3:Questions to ponder:1) What can you add to 3 to get the additive identity as your answer?2) What can you add to -4 to get the additive identity as your answer?3) What can you add to x to get the additive identity as your answer?4) Write a general statement and equation that describes this property.5) Why do you think this is an important idea to know about addition?Provocateur: A question you might ask for discussion is, why is this property important for the understanding of addition of signed numbers?Activity 4:Reflector: Before teaching this lesson think about, why is the multiplicative identity an important idea for students to think about?Questions to Ponder1) What can you multiply 3 by to get the multiplicative identity element?Coach: As a heads up, students often struggle with making sense of multiplying to get 1. How can you help them make sense of the notion of the getting to the identity?2) What can you multiply -4 by to get the multiplicative identity element?3) What can you multiply x by to get the multiplicative identity element?4) Write a general statement and equation that describes this property.5) Why do you think this is an important idea to know about multiplication?Closing Activity: Journal Writing:Why do you think it is important to understand these different properties?Learner: Collect these writing pieces from your students and see what connections they were able to make within the material in the lesson as well as to other ideas in mathematics that they may have experienced before. Decision Maker: If there are a few pieces that stand out to you from what you collected you could use those at the beginning of the next day’s lesson.Unit 2: Equality, Inequality & Problem SolvingLesson 2: Are both methods equivalent?Student VersionName_______________________Date________________________Opening Activity19050128270You and your friend Shawn are selling boxes of candy on the bus/subway. You begin your speeches by saying, “I’m not going to lie to you. I’m not selling candy for my team, my church or any other organization. I’m selling this for me, to keep me off the street.” At the end of the week, your plan is to share the profits equally. For each box of candy sold, you and Shawn make $7. You sold 18 boxes and Shawn sold 23 boxes.Shawn’s method:Shawn chose to calculate the amount you and he will earn by first adding number of boxes sold and then multiplying 7.Your method:You figured out your own earnings before you figured out Shawn’s earnings. Then you added your earnings and Shawn’s earnings pare your results with Shawn’s. What do you notice?Which method makes more sense to you? Why does it make more sense?1) Let m stand for the number of boxes you sold, s stand for the number Shawn sold and let c stand for the amount you earn on each box.Write an algebraic expression showing the method that Shawn used.Write an algebraic expression representing the method that you used.Write an equation that includes the information from both of your methods.2) Experiment with some other numbers to see if your equation holds true.Let:m = 4, s = + 6, and c = 9Let:m = 3, s = -2, and c = -4Do you think this will hold true for any type of numbers we use? What makes you think this?3) Do you think this property is true for multiplication over subtraction? Try several examples and report your conclusion.Activity 2:1) Pick a number, any number. What would you add to this number so that the result does not change?Try this with a couple of other numbers.2a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How do you know?2b) Write a general equation. Use x represent to the numbers you and your classmates chose.3) Pick a number, any number. What would you multiply this number by such that the result does not change?Try this with a couple of other numbers.4a) Write an equation with your number that shows this to be true. Is this true for any number you could choose? How do you know?4b) Write a general equation. Use x represent to the numbers you and your classmates chose.Activity 3:Questions to ponder1) What can you add to 3 to get the additive identity as your answer?2) What can you add to -4 to get the additive identity as your answer?3) What can you add to x to get the additive identity as your answer?4) Write a general statement and equation that describes this property.5) Why do you think this is an important idea to know about addition?Activity 4:Questions to ponder1) What can you multiply 3 by to get the multiplicative identity element?2) What can you multiply -4 by to get the multiplicative identity element?3) What can you multiply to x to get the multiplicative identity element?4) Write a general statement and equation that describes this property.5) Why do you think this is an important idea to know about multiplication?Closing Activity: Journal Writing:Why do you think it is important to understand these different properties?Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 3: Two Coupons MysteryTeachers GuideDecision Maker: Today’s lesson will be a group performance task. That means you will form groups of two or three students and ask them to work on the problem as a group without your support. You want to make sure they understand the problem, so you can answer clarifying questions, but they must think about this with each other. Reflector: When there are 15 minutes left in the class bring the class together to begin to discuss the problem. It should not be about answering all the questions, but around student thinking and process. This problem really focuses on whether order makes a difference: Is discounting the 20% first different from discounting the $5 first?3025866120659100570320127027200Decision Maker/Artist: If you think coupons and calculating with percentages will get in the way of the task, you can try this. Look for various different coupons and create an opening activity in which you ask students what these are, what function do they serve? How might one use them? Allow students to lead the conversation and work out the mathematics involved. Discussing this first will allow students to focus on the real purpose of the task, to see if order makes a difference depending on the operations being used.298150723685500Artist: Act it out! Give students two coupons and a $50 price tag. Does changing the order of the coupons change the cost? This is especially wonderful for ELLs so that the confusion isn’t in deciphering language.58388252540Group Performance Task: Mystery of the Two CouponsStephanie and Julio go jeans shopping. They each have a 20% off coupon and a $5 off coupon. They each found a pair of jeans that they really like for $50. When they compared receipts after going through the register, they saw that Stephanie had paid less than Julio even though they had the same coupons and they each paid 10% in tax. What happened at the register that made the difference? Support your answer with mathematical evidence.KnowNeed to KnowNext StepsUnit 2: Equality, Inequality & Problem SolvingLesson 3: Two Coupons MysteryStudent VersionNames _____________________________________________________________________________Date ________________________5791200179705Group Performance Task: Mystery of the Two CouponsStephanie and Julio go jeans shopping. They each have a 20% off coupon and a $5 off coupon. They each found a pair of jeans that they really like for $50. When they compared receipts after going through the register, they saw that Stephanie had paid less than Julio even though they had the same coupons and they each paid 10% in tax. What happened at the register that made the difference? Support your answer with mathematical evidence.KnowNeed to KnowNext StepsUnit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 4: Four Laws of EqualityTeachers GuideCoach: Our goal throughout the rest of the unit is twofold. First, we want students to have a meaningful understanding of what equality is and how it is linked to solving. This means having the understanding that as we make changes through the operations, the two sides remain equal to each other. In other words, although we change the way each side looks, they still remain the same in their equality. This powerful point often gets lost in student thinking. Secondly, we want students to develop flexibility in their ability to approach solving equations. Developing both of these in our students takes time. All too often, teachers rush students to an efficient means of solving an equation rather than letting them work inefficiently for a while. Remember in the mess of inefficiency is where the long-term understandings will be forged, so resist the urge to correct an inefficient method and stay focused on developing your students’ understanding of equality and flexibility. If you rush your students towards efficiency you will have enabled them to repeat your methods, but will have stunted their ability to think flexibly about solving.Opening Activity:Suppose we write: 3 + 8 = 4 + 7Is this true? How do you know?What does the symbol = mean, without using the word equal?Decision Maker: One of the first things is to have a conversation about how to refer to each side of the equal sign and how you are going to talk about it as a class. This may sound trivial, as it seems intuitive to refer to each side of the equal sign using directional language such as left and right, but developing this language with the students gives you the opportunity to get everyone in the room on the same page. Once the language about how you are going to talk about the contents on each side of the equal sign is established, it’s important to explore what the students think is the meaning of the equal sign. Student understanding has been developed over many years and often contains misconceptions or limited understanding. You’ll learn a lot about their concept of mathematical equality simply by asking. “What would a good answer be to: What is the meaning of an equal sign?”We want to prepare our students for solving equations and one can only understand the process of solving if one understands the full meaning of equality.Transition to next activity -- Provocateur: Suppose we started with an equation and wanted to use addition to create another equivalent equation. How would we do this?Reflector: The rest of this lesson is focused on the unique power that equality provides mathematics. This lesson ends with a big question that is important for you to give some thoughtful reflection to prior to teaching this lesson:Why is equality important to mathematics and to us as we explore the world of mathematics?Activity 2:Decision Maker: There are several ways to facilitate this activity; a few of them are outlined below. The goal in the remaining activities is to see the power that equality provides us in working with equations and to give a name to how (using the laws of equality) we create new equivalent equations.Artist: Knowing your students, feel free to modify this activity so that it’s sure to engage all of your students.Facilitation Option 1: Students work in small groups and whole class discussions take place after each part of this Activity.(A) Now look at:3 + 9 + 4 = 4 + 8 + 4Is this still equal? Why?Try this with some other numbers, can you add any number to both sides and still maintain equality? Explain.Can you come with a general statement to describe this law?Decision Maker: Have students share their statements. Pick one that best describes this property of equality, write it on chart paper and place it on the wall. Name it after the student who developed it, e.g. Jose’s Law.Provocateur: Through all of these examples the focus is not on getting through the examples, but seeing the maintenance of equality as well as developing a sense of flexibility in the students. Have fun with it. Include several nontraditional examples with really big or really small numbers, or exponents. For instance, with multiplication you could do 1,000,000(3+9) = 1,000,000(4+8).Decision Maker/Reflector: Why would you want to give your students exercises like the type we suggest? What would you learn about your students’ understanding?Your students will pick the options they are comfortable with, but be sure to give them a larger perspective on the power of equality and the flexibility we do have when maintaining equality. You’ll need to exhibit the kind of flexibility you want to see in your students’ understanding and you can help build some of that by encouraging them to think outside of what’s comfortable or obvious.What does this say about mathematics? Why is this important to mathematics and to us?(B) Now if we look at 3 + 9 - 9 = 4 + 8 - 9Is this still equal?Why?Try this with some other numbers, can you subtract any number from both sides and still maintain equality? Explain.Can you come with a general statement to describe this law?Decision Maker: Have students share their statements. Pick one that best describes this property of equality, write on chart paper and place on the wall. Name it after the student who developed it, e.g. Nadia’s Law.(C) Now if we look at5(3 + 9) = 5(4 + 8)Is this still equal?Why?Try this with some other numbers: can you multiply any number by both sides and still maintain equality? Explain.Would this work for a multiplication by a negative number? Give an example.Can you come with a general statement to describe this law?Decision Maker: Have students share their statements. Pick one that best describes this property of equality, write on chart paper and place on the wall. Name it after the student who developed it, e.g. Richard’s Law.(D) Now if we look at(3 + 9)/2 = (4 + 8)/2Is this still equal?Why?Try this with some other numbers, can you divide by any number on both sides and still maintain equality? Explain.Would this work for division by a negative number? Give an example.Can you come with a general statement to describe this law?Decision Maker: Have students share their statements. Pick one that best describes this property of equality, write on chart paper and place on the wall. Name it after the student who developed it, e.g. Jackie’s Law.Facilitation Option 2: In small groups (3s or 4s) students create equivalent equations using each of the basic operations. Provide each group with a sheet that has the name of the operation at the top; you could use colored paper to help create a unique visual effect when we bring all the examples back together. Towards the end of this activity you are going to have students put up their sheets full of equivalent equations on a poster for the corresponding operation.AdditionCreate as many equivalent equations as you can with 3+9=4+8 as your starting equation. MultiplicationCreate as many equivalent equations as you can with 3+9=4+8 as your starting equation. DivisionCreate as many equivalent equations as you can with 3+9=4+8 as your starting equation. SubtractionCreate as many equivalent equations as you can with 3+9=4+8 as your starting equation. Students then create, using the operation identified, as many equivalent equations as they can when given as their initial equation 3+9 = 4+8.Your role as Coach, and Questioner will be important in this process.Provocateur: Encourage them to be creative, use big numbers, radicals, rational numbers, etc.Decision Maker: As you are walking around identify some of the more creative ones that students made and ask them to post them on the respective poster for the whole class to see.When each group has created several new equivalent equations have them put up their sheets one the poster for that operation. This creates a visual representation of the many equivalent equations that can be created using the idea of equality in conjunction with each of the basic operations. Reflector: Pose questions like the following to the class to move towards identifying the four laws of equality: How many of these equivalent equations could we have created? What’s common to how all of these equivalent equations were created? Facilitation Option 3: In small groups (3s or 4s) students create equivalent equations using ONE of the basic operations, but every group starts with the same equation 3+9 = 4+8. Also pose the following questions to the students: (1) How were you able to maintain equality in each equivalent equation? (2) Do you think this would work using any kind of number? How do you know? (3) Based on the work you did with your operation, how do you think creating equivalent equations with the other operations works?Coach: As you walk around the room identify a group to share for each of the operations.Provocateur: Encourage them to be creative, use big numbers, radicals, rational numbers, etc.Have students share out and give names to the properties discussed.Activity 3:(A) Let’s look at a few more equalities.Suppose:4+10 = 3 + 11Now look at the following:4 + 10 -10 = 3 + 11 -10We know from a previous law that this is still equal.Look at the left side of the equation after you performed the operations. Notice that we’re left with 4, the first number in the equality. Why did that happen? Can you recall the idea we learned that predicts this?Coach: We want students to connect that we use the additive inverse which results in a sum of zero, which is the additive identity, to get the 4 alone.(B) If I had8 – 5 = 2 + 1What can I do to both sides of the equation so that each side equals 8?Let’s do another one. If I had -7 + 11 = 1 + 3What would you do to have both sides of the equation equal 11?Coach: We want students to come to the conclusion that you have to add the additive inverse to each side of the equation to get a particular number in the equation by itself.Now you should create an equality similar to the previous two and decide which number on either side you want to isolate (remain by itself). Be ready to present yours to the class.Suppose we have:5(6) = 30What can you do to isolate the 6? Explain.Now if we have:-2(-4) = 8What can you do to isolate the -4? Explain.Coach: We want students to come to the conclusion that you have to multiply each side of the equation by the multiplicative inverse to get 1, the multiplicative identity, to get a particular number in the equation by itself. (We are not canceling out!) We use the inverse and identity to maintain equality.Make a general statement that will help us to isolate a particular number in an equation that involves multiplication.Now let’s look at one final equation:105=2What can you do to isolate the 10? ExplainMake a general statement that will help us to isolate a particular number in an equation that involves division.Closing Activity:Journal Writing: Why is equality important to mathematics and to us as we explore the world of mathematics?Practice: Isolate the darkened number in each of the following equations, using the ideas you have just learned. Show all your steps and prove that you’ve maintained equality.4 + 12 = 9 + 7-6 + 9 = 2 + 18 – 4 = 15 – 116 + 9 = 20 – 58(4) = 32-5(3) = -15246=4246=4Reflector: As we transition into solving be sure to communicate to the students that their work that allowed them to arrive at their answer and their explanation of that process is just as, if not more, important than the answer itself. You will, if you haven’t already in your career, have students who push back on having to show work or ask for justification for why they should show work. It is very important that your reflection on this question goes beyond, ‘because I said so’. That response is not sufficient to anyone no matter what the age. This conversation is an opportunity to help expand your students’ vision of mathematics. Help them to understand that mathematics is a human endeavor, thus a major component of mathematics is the ability to communicate what we are doing and why we are doing it. Additionally, just like text, emails, and voicemails, enables us to communicate when we aren’t around, showing our work enables us to communicate to each other what we are doing and how we are thinking about something when the reader is not around. Any time you have the opportunity to humanize and expand your students’ perceptions of mathematics take advantage of it. Hopefully, as a result of taking advantage of opportunities like this, your students will leave your class with a more robust view of mathematics than the one they came in with.Unit 2: Equality, Inequality & Problem SolvingStudent Version for Facilitation Option 1:Lesson 4: Four Laws of EqualityStudent VersionName_______________________Date________________________Opening Activity:Suppose we write: 3 + 8 = 4 + 7Is it a true statement? How do you know? Define the symbol = without using the word equal.(A) Now look at:3 + 9 + 4 = 4 + 8 + 4Is this still equal? Why?Try this with some other numbers, can you add any number to both sides and still maintain equality? Explain.Can you come with a general statement to describe this law?(B) Now if we look at 3 + 9 - 9 = 4 + 8 - 9Is this still equal?Why?Try this with some other numbers, can you subtract any number to both sides and still maintain equality? Explain.Can you come with a general statement to describe this law?(C) Now if we look at5(3 + 9) = 5(4 + 8)Is this still equal?Why?Try this with some other numbers, can you multiply any number to both sides and still maintain equality? Explain.Would this work for a multiplication by a negative number? Give an example.Can you come with a general statement to describe this law?(D) Now if we look at(3 + 9)/2 = (4 + 8)/2Is this still equal?Why?Try this with some other numbers, can you divide by any number on both sides and still maintain equality? Explain.Would this work for division by a negative number? Give an example.Can you come with a general statement to describe this law?(E) Let’s look at a few more equalities.Suppose:4+10 = 3 + 11Now look at the following:4 + 10 -10 = 3 + 11 -10We know from a previous law that this is still equal.Look at the left side of the equation after you performed the operations. Notice that we’re left with 4, the first number in the equality. Why did that happen? Can you recall the idea we learned that predicts this?(F) If I had8 – 5 = 2 + 1What can I do to both sides of the equation so that each side equals 8?Let’s do another one. If I had -7 + 11 = 1 + 3What would you do to have both sides of the equation equal 11?Now you should create equality similar to the previous two and decide which number on either side you want to isolate (remain by itself). Be ready to present yours to the class.(G) Suppose we have:5(6) = 30What can you do to isolate the 6? Explain.Now if we have:-2(-4) = 8What can you do to isolate the -4? Explain.Make a general statement that will help us to isolate a particular number in an equation that involves multiplication.(H) Now let’s look at one final equation:105=2What can you do to isolate the 10? ExplainClosing Activity/Exit Ticket: Make a general statement that will help us to isolate a particular number in an equation that involves division.Unit 2: Equality, Inequality & Problem SolvingStudent Version for Facilitation Option 2:Lesson 4: Four Laws of EqualityStudent VersionName_______________________Date________________________Practice: Isolate the darkened number in each of the following equations, using the ideas you have just learned. Show all your steps and prove that you’ve maintained equality.4 + 12 = 9 + 7-6 + 9 = 2 + 18 – 4 = 15 – 116 + 9 = 20 – 58(4) = 32-5(3) = -15246=4246=4Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 5: What is the meaning of equality?Teacher GuideLearner: In Lesson 4 students worked to understand the power of equality and, in particular, four manifestations of equality with the basic operations. In Lesson 5 the idea of equality is being expanded to include equations with variables. The focus on this lesson is on how equality can be created when a variable is present in an equation. Solving for the variable is not the main emphasis in this lesson. Lesson 6 will get more into that. Opening Activity: What would be the value of x to make this statement true?Learner: In the Opening Activity you will have an opportunity to see if and how students use their experience from Lesson 4 to make sense of equations with variables.Suppose you have the statements:x + 3 = 317 (x) = 21 Using any method you can, find out what the value of x must be in each statement to maintain equality. How did you do it?Coach: Have students share their responses, but remember we are working on flexibility, not efficiency right now. If a student wants to add 7 to both sides, because they’d prefer to work with 10 on the side with the x, let it happen. Let what the other students did inform how that student’s perspective on solving grows.Leaner: Did any of the students use the ideas from yesterday? If so have them talk about what connection they made from the work yesterday to this activity. It will be helpful for other students to hear how their peers made that connection. Decision Maker: If the ideas from Lesson 4 aren’t present in anyone’s thinking consider asking the class, “How can we use the ideas of equality to help us here?” Then giving the students some time to work that out.Decision Maker: If you used facilitation option 1 in lesson 4, be sure to use the language created in that lesson. For example, “I isolated the x by adding -3 to both sides of the equation, which was Jose’s Law.”In Lesson 4, we worked with isolating particular numbers in the equation. Today we are working with isolating a variable in the equation. What similarities and differences have you seen so far? Be prepared to share your thoughts with the class.Reflector: Ask students to share-out their thinking on the similarities and differences. The fact that they have used the laws they developed yesterday is testimony that variables shouldn’t behave differently than the numbers they are replacing. We want students to feel a certain sense of familiarity with the step they’ve made in working with variables now. Here again we are bridging from arithmetic and numbers to algebra and variables.Activity 2: Always, Sometimes, NeverReflector: The equal sign takes on different meanings depending on how the mathematical statement is presented. For example, if you are asking is this statement true? x+1= x+1 . The answer you would have to say is no, or sometimes, because it is only true when x equals 0. However, if you asked is 2(x+3) = 2x + 6 true? You would have to say yes, because any value of x would make this true. So we are asking students to get deeper about the meaning of equality based on the mathematical statement.Note to teacher: Students will be given the following equations and they have to figure out the value(s) of x in each whichever way they want.For each of the following statements, determine what value(s) of x would make the statement true.2(x+3) = 2x + 6x + 5 = 11x + 3 = x + 2Provocateur: This should lead to a discussion about equations in which the value of x can be any real number, one real number, or that there are no values of x that would make the statement true. Share with them the term “True Equations”, which means that the solution to the equation can be any real number. This term only applies to the first example. Provocateur: What about these equations told us that they would be always true, sometimes true or never true for values of x? With this question you want to help students focus on the characteristics of each of these descriptors (always, sometimes, never) of an equation.Activity 3:Create three equations: one equation that’s always true, one that’s sometimes true, and one that’s never true.Learner: During this activity look at and listen in on the students’ process for creating these equations. Their discussions and work will reveal what sense they have made from the discussion about variables and equality.Decision Maker/Questioner: How will you respond to students struggling to create these equations? What will you draw their attention to? Focus your comments on equality and the qualities of each of the equations the students worked with in Activity 2.Challenge: If 3 froogles = 2 froogles + 4 moogles,what does one froogle equal?How did you use the laws of equality and the properties to determine that value?Closing Activity:As best you can, write about the meaning of equality using the experiences you’ve had in the last couple of lessons.Learner: Facilitate a share out of the students’ reflections. Here again you are listening for what sense students are making out of the idea of equality. Are they comfortable with how the operations can create other equivalent equations? What connections have they made between equations, variables, and equality?Homework:Always, Sometimes, Never revisited:For each of the following equations, determine whether there is one solution, no solution, or an infinite amount of solutions. Explain your reasoning.3x = 2xx + 2 = x + 32(x +4) = 2x + 82x + 4 = 2(x-6)Unit 2: Equality, Inequality & Problem SolvingLesson 5: What is the meaning of equality?Student VersionName_______________________Date________________________Opening Activity:Suppose you have the statements:x + 3 = 317 (x) = 21 Using any method you can, find out what the value of x must be in each statement to maintain equality. How did you do it?In Lesson 4, we worked with isolating particular numbers in the equation. Today we are working with isolating a variable in the equation. What similarities and differences have you seen so far? Be prepared to share your thoughts with the class.Activity 2:For each of the following statements, determine what value(s) of x would make the statement true.2(x+3) = 2x + 6x + 5 = 11x + 3 = x + 2Activity 3:Create three equations: one equation that’s always true, one that’s sometimes true, and one that’s never true.Challenge: If 3 googles = 2 googles + 4 moogles,what does one google equal?How did you use the laws of equality and the properties to determine that value?Closing Activity:Journal writing: As best you can, write about the meaning of equality using the experiences you’ve had in the last couple of lessons.Homework:Always, Sometimes, Never revisited:For each of the following equations, determine whether there is one solution, no solution, or an infinite amount of solutions. Explain your reasoning.3x = 2xX + 2 = x + 32(x +4) = 2x + 82x + 4 = 2(x-6)Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 6: Solving Equations, Maintaining Equivalence & JustificationLearner: What do we mean by the word “solve”? What do we want students to understand when they hear the word “solve”? We want students to understand that when one is solving, the ideas of solving are based on the Laws of Equality. We will talk about the word “solution” throughout this unit and the different meanings it takes on based on the type of equation or inequality one is working with.Note to Teacher: Students traditionally struggle with solving equations because they perceive it as a set of rules and procedures that must be followed to arrive at “the answer”. It is true that a procedural understanding is involved and necessary for carrying out calculations; however, the conceptual understanding that must exist is equally, if not more, important. A student must be able to examine the structure of an equation to determine the appropriate procedures that must be employed to solve it. To be a flexible successful solver of equations the deep conceptual understanding of equality will lead to a natural sense of how to proceed. Here* are several researchers’ takes on the relationship between conceptual and procedural understanding in solving equations.* to Teacher: Another important idea of this lesson is for students to develop the understanding that when we manipulate the equations through the process of solving them, the original equation and new equations created are equivalent. Opening Activity: What does it mean to “solve for x”? Learner: The question is often stated more as a command: Solve for x. When phrased as a command, it is still considered a question – “What value (or values) exist that will make the statement of equality hold true?” When we say that we have “solved for x,” what we are really saying is that we have gone through a process guided by the laws of equality and properties of arithmetic to find a value(s) for “x” (unknown) that makes the equation true. Learner: Are students demonstrating the understanding that an equation is a statement of equality between two expressions? Activity 2: Learner: You will give the students 3x+3 = 12 with the purpose of seeing what they can do and learning about how they can explain what they did and why they did it.Solve: 3x + 3 = 12What did you do?Why did you do it?What laws and properties did you use in your process?Decision Maker: The important idea here is focusing on What did you do? & Why did you do it? The format answering these questions takes is not as important as the content of the answer. Decide how you want to talk about this with your students. One way this could be done is given below and you can decide whether or not you want to show them this approach. 3x + 3 = 12Given -3 -3Equality law for addition or adding the additive inverse 3x = 9 3x3=93Equality law of division or multiplying by the multiplicative inverse x = 3Coach: This may be too big of a jump for some students, so work with a specific example first then include the variable. For example have the student or students work with 3(4) + 6 = 18 where they isolate the 4 then have them work with 3(x) + 6 = 18. Here they can see that they took the same steps with or without the variable being present.Activity 3: Solve for x: 4(x + 2) + 6 = 6x – 4 What did you do?Why did you do it?What laws and properties did you use in your process?Coach/Provocateur: Show all the steps on the board and pose the question, what can we say about the relationship between the original equation and new equations created through the process of solving for x? This is a crucial idea for a student’s ability to have flexibility because he/she is making sense of the equivalence between all equations in the process of solving equations. We can think of the equations that result from manipulations we do as we solve for x as a string of equivalent equations. AB The distributive property: 4(x + 1) = 4x + 4, and statement A is equivalent to statement B. B C The associative property: 4x + 4 + 6 = 4x + 10, and statement B is equivalent to statement C.CD The additive inverse property and laws of equality: 4x + 10 – 10 = 4x and 6x + 14 – 10 = 6x + 4, and statement C is equivalent to statement D.And so on…..Provocateur: If I gave you statement D and asked you to solve it, what result would you get?3364278175885433552801273223A. 4x+1+6=6x+14B. 4x+4+6=6x+14C. 4x+10=6x+14 -10 -10D. 4x= 6x+4 -6x -6xE. -2x=4-2x-2=4-2F. x= -2Closing Activity: Further understanding of the laws of equalityx + 2 = 6x + 6 = 102x + 12 = 205x + 12 = 3x + 20What is changing and what is staying the same?Learner: This is an opportunity to see what your students understand about the relationship of equivalent equations in the process of solving an equation.Provocateur: Given x + 2 = 6, we want students to see that a new, but equivalent, equation results from adding x to both sides of the equation. Furthermore, we can multiply both sides of the equation by 2 and result in a new equivalent equation, and so on…Provocateur: Since we can call these (x + 2 = 6 and x + 6 = 10 and 2x + 12 = 20 and 5x + 12 = 3x + 20) equivalent equations this question will help your students begin to develop more nuance in their understanding of equality. Also, using the students’ work for solving the two equations will help the students see that when you maintain equality you will always be creating an equivalent equation. Coach: Now the big thing you want the students to see is that if you solve each equation separately you will get the same solution. In all four cases x = 4. So get your students to solve these two equations and talk about what happened.Exit Ticket: How do you see the idea of equality showing up in the solving process? Homework: Decision Maker: Here is a homework you can give to students to see if they understand what it means for two equations to be equivalent. In Pair C, students with basic understanding might think that only 2x = 0 and 3x = x are equivalent (because x was added to both sides of 2x = 0). We are looking for students to see that these are also equivalent because they both have the same solution.Part 1:Examine each pair of equations. Determine whether each pair represents a set of equivalent equations. Explain your reasoning.Pair A: x = 1 and 2x = 2Pair B: x = 1 and x = 2Pair C: 2x = 0 and 3x = 0Pair D: 2x = 0 and 3x = xPart 2:Create a new equation that will be equivalent to 3x + 2 = 4x + 3. How can you be sure that they are equivalent?Part 3: Regents exam practice When solving the equation 43x+2-9=8x+7, Emily wrote 43x+2=8x+16 as her first step. Which property justifies Emily’s first step? Addition property of equalityCommutative property of additionMultiplication property of equalityDistributive property of multiplication over additionContinue solving Emily’s equation.Unit 2: Equality, Inequality & Problem SolvingLesson 6: Solving Equations with JustificationStudent VersionName_______________________Date________________________Opening Activity: What does it mean to “solve for x”? Activity 2:Solve: 3x + 3 = 12What did you do?Why did you do it?What laws and properties did you use in your process?Activity 3: Solve for x: 4(x + 2) + 6 = 6x – 4 What did you do?Why did you do it?What laws and properties did you use in your process?Closing Activity: Further understanding of the laws of equalityx + 2 = 6x + 6 = 102x + 12 = 205x + 12 = 3x + 20What is changing and what is staying the same?Exit Ticket: How do you see the idea of equality showing up in the solving process? Homework: Part 1:Examine each pair of equations. Determine whether each pair represents a set of equivalent equations. Explain your reasoning.Pair A: x = 1 and 2x = 2Pair B: x = 1 and x = 2Pair C: 2x = 0 and 3x = 0Pair D: 2x = 0 and 3x = xPart 2:Create a new equation that will be equivalent to 3x + 2 = 4x + 3. How can you be sure that they are equivalent?Part 3: Regents exam practiceWhile solving the equation 43x+2-9=8x+7, Emily wrote 43x+2=8x+16 as her first step. Which property justifies Emily’s first step? Addition property of equalityCommutative property of additionMultiplication property of equalityDistributive property of multiplication over additionContinue solving Emily’s equation.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 7: Practice with equations, work with errorsDecision Maker: You need to gauge where your students are with equations and how much practice you need to give them. Make sure they can talk about what they did, why they did it, and how they maintained equality by creating new equivalent equations.The Closing Activity included in this lesson is important. The students’ writing will be used in the Opening Activity in Lesson 8:Journal writing: Describe, in your own words, what you do when you are asked to solve an equation. Be sure to focus on the process, what you are doing, why you are doing it, and how you see equality being maintained throughout your solving process. Feel free to use an example, if that will help you formulate your thoughts.Learner: What are you seeing in your students’ writing? Does is sound more like rigid procedure or are they developing flexibility? Remember we want to develop students who can think flexibly about how they approach solving equations. The opening activity for the next lesson is based on the students’ responses to the prompts. Be sure to collect them and read through them. Choose a few different pieces: one that exemplifies flexibility in the approach to solving and one that is rigid in the approach to solving. Another contrast that could be set up is between a student who is making meaningful connections between maintaining equality and the properties and another student who is not doing that.Decision Maker: How do you want to facilitate this lesson? Do you want to have students work on one at a time then have students share, including students who made interesting errors? You also might have them work on a few at a time so you can sit and observe what is going on with each student and learn about where students are struggling. Your goal is to learn about how your students are thinking when they try to solve different equationsHere are some equations you might include in this lesson:a) -3x – 2 = -11 b) -6 + 5x = 8 – 2x c) 12 – 3x = 15d) 8 – 4x = 2x e) 3(2x – 6) = -4( 8 + x) f) 2x - 6 = 5x 3Activity 2: Do decimals and fractions change your approach to solving an equation?For example, is there a difference in the process you would use to solve each of the following equations?3x + 5 = 152x + 2.3 = 4.712+5x=112 Decision Maker: Here are multiple examples. Use them depending on the information you want to gather about your students.g) 5x – (3x + 2) = 1h) 4-13x=23x i) 600x = 200x + 40j) 1250x=3k) 0.63x + 7 = 0.84xl) 0.5(3x + 2) = 0.25m) 5(1x + 3) = 25Exit Slip: Do the laws of equality hold true for all types of equations, including those that have fractions and decimals? Create an example to support your thinking.Journal writing: Describe, in your own words, what you do when you are asked to solve an equation. Be sure to focus on the process, what you are doing, why you are doing it, and how you see equality being maintained throughout your solving process. Use examples if they will help you formulate your thoughts.For example, is there a difference in the process you would use to solve each of the following equations?3x + 5 = 152x + 2.3 = 4.712x+5=112x Decision Maker: The last example might seem daunting for students. You may offer the following as a scaffold and students can use it to answer the journal prompt 12x+5= 112x -12x -12x5= 102x5=5x 55 5x5x=5Homework: a) Solve for x.13x+5+ 23x+5=7b) Try to solve the equation in a different way.-Unit 2: Equality, Inequality & Problem SolvingLesson 7: Practice with equations, work with errorsStudent VersionName_______________________Date________________________a) -3x – 2 = -11 b) -6 + 5x = 8 – 2x c) 12 – 3x = 15d) 8 – 4x = 2x e) 3(2x – 6) = -4( 8 + x) f) 2x - 6 = 5x 3Exit Slip: Do the laws of equality hold true for all types of equations, including those that have fractions and decimals? Create an example to support your thinking.Journal writing: Describe, in your own words, what you do when you are asked to solve an equation. Be sure to focus on the process, what you are doing, why you are doing it, and how you see equality being maintained throughout your solving process. Use examples if they will help you formulate your thoughts.For example, is there a difference in the process you would use to solve each of the following equations?3x + 5 = 152x + 2.3 = 4.712x+5=112x Homework: a) Solve for x.13x+5+ 23x+5=7b) Try to solve the equation in a different way.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 8: Am I a flexible Thinker: 4 VignettesTeachers GuideOpening Activity: Student Reflections on SolvingDecision Maker: As mentioned in the note to the teacher at the end of the last lesson the Opening Activity is based on the students’ journal writing from the end of Lesson 7. So be sure to read their responses keeping an eye out for ways to setup a contrast between the ideas communicated in the journal entry. You could setup a contrast between a flexible and non-flexible solver or between a student who is making the connections to maintaining equality and the properties and one who is not. The goal in approaching the opening activity in this way is to give students the opportunity to hear how their classmates are talking about it as well as hear other ways to think about solving. Be sure to make the student work anonymous to avoid any unintentional discomfort.Coach: Students need exposure to tasks specifically designed to develop their flexibility in solving equations. For example, instead of using the equation 3(x + 1) = 22, we might change the problem to 3(x + 1) = 21, to increase the likelihood that students become aware that they could divide both sides by 3 first and actually use this strategy. In addition asking students to solve the same problem in more than one way and to identify which method was more efficient/preferred by them helps to develop both flexibility and efficiency in solving equations. Activity Two: 4 VignettesDecision Maker: We are going to use 4 short vignettes to help stretch the students’ ability to think flexibly as problem solvers. The last one will be an individual performance task (be sure to allocate time accordingly). After each of the first three vignettes you should facilitate a discussion about the vignette.Decision Maker: Feel free to edit the way the work is shown to coincide with how you want work to be shown in your class.Vignette 1: Done two ways, but who’s correct?Julio and Lilly are both in your group. You all solved the same equation. When Julio looked at Lilly’s work he noticed his work was different. They both got the same answer. Julio says his way is right, but Lilly just thinks he got lucky.Lilly’s WorkJulio’s WorkHelp them figure out how to make sense of this situation.Learner: Let the students wrestle with their ideas. Facilitate a discussion. We want to redefine correctness around the solution and understanding multiple ways to the solution rather than correctness being only ascribed to one way of solving for a variable. Provocateur: A follow-up question could be: When would Julio’s approach be less efficient? You might want to throw this problem at kids: 3x + 9 = 5x – 12. Would there be a problem here? When would I want to divide first, when wouldn’t I want to? It’s not that you can’t divide by 3 first, it would just become a little messier.Vignette 2: Can we get a zero while solving?Rafael and Roman, the twins in the class, solved this equation differently.Rafael did the work shown below and got stuck. He’s convinced his work is correct to this point, but isn’t sure how to finish.Roman believes he’s correct and has done this work so far:The other students in the class are yelling, “You’re both wrong!”, but the twins are determined and insist that they maintained equality.Help figure out what’s happening here. What are the issues? Did they both maintain equality? How do you know?Provocateur: We want to have a discussion: can one side of the equal sign equal zero? Why? We also want to discuss meaning of equality and the role the equal sign plays in that definition. What happens if you move the equal sign to a different location in the equation? Remind students that the equal sign is not a separator but rather a statement of equivalence.Vignette 3: What went wrong?Natasha needs your help. She did the following work:Check:Natasha said, “What did I do wrong? I thought I could divide first. Was I right or wrong?”Help Natasha sort out her situation.Provocateur: When we add, it seems like we are only adding to the constant term, but when we divide we are dividing every term. Why? If there is a confusion about this, think about these two problems: (6+ 9) + 3 and (6 + 9)/3 to help describe the behavior. Vignette 4: Individual Performance Task – Jose’s EquationJose says to Jill, “I can solve for the variable in this equation in at least 4 different ways.” Jill says, “No, you can’t.”Jose’s equation is: -4(x+2) = 8x +16Prove if Jose is right or wrong.Reflector: What have you learned about your students from these vignettes? What will be your next steps in instruction? You want to push the dialogue about why acknowledging a flexibility in thinking about solving equations is important; Flexibility is the understanding of what we can and cannot do when solving an equation; our understanding of the properties of our number system and the laws of equality create the flexibility. Decision Maker/Learner: you want your students to reflect on this experience. If we want them to develop flexibility, we must have conversations about what it means to be flexible thinkers. What can you learn about your students through these reflection questions?Journal Writing/Exit Slip: Choose one of the following questions to answer:How do the laws of equality allow us to be flexible with our approach to solving equations? Cite specific examples from the vignettes. ORWhy is flexible thinking necessary in solving equations? Can all equations be solved in exactly the same way?Homework Decision Maker: It may be advantageous at this point to provide students with additional experience through homework, but you’ll need to gauge and decide what, how much, and the outcomes for which you are looking.Decision Maker: Here is a sample homework assignment in which students examine errort in student work. Why would you assign this type oftask to your students?Student’s WorkHow is the student thinking incorrectlyWhat would you say to the student to help him understand?49232275581005160747967300587325108107006268115302600Unit 2: Equality, Inequality & Problem SolvingLesson 8: Flexibility with EquationsStudent VersionName_______________________Date________________________Vignette 1: Done two ways, but who’s correct?Julio and Lilly are both in your group. You all solved the same equation. When Julio looked at Lilly’s work he noticed his work was different. They both got the same answer. Julio says his way is right, but Lilly just thinks he got lucky.Lilly’s WorkJulio’s WorkHelp them figure out how to make sense of this situation.Vignette 2: Can we get a zero while solving?Rafael and Roman, the twins in the class, solved this equation differently.Rafael did the work shown below and got stuck. He’s convinced his work is correct to this point, but isn’t sure how to finish.Roman believes he’s correct and has done this work so far:The other students in the class are yelling, “You’re both wrong!”, but the twins are determined and insist that they maintained equality.Help figure out what’s happening here. What are the issues? Did they both maintain equality? How do you know?Unit 2: Equality, Inequality & Problem SolvingVignette 3: What went wrong?Natasha needs your help. She did the following work:Check:Natasha said, “What did I do wrong? I thought I could divide first. Was I right or wrong?”Help Natasha sort out her situation.Unit 2: Equality, Inequality & Problem SolvingVignette 4: Individual Performance Task – Jose’s EquationJose says to Jill, “I can solve for the variable in this equation in at least 4 different ways.” Jill says, “No, you can’t.”Jose’s equation is: -4(x+2) = 8x +16Prove if Jose is right or wrong.Journal Writing/Exit Slip: Choose one of the following questions to answer:How do the laws of equality allow us to be flexible with our approach to solving equations? Cite specific examples from the vignettes. ORWhy is flexible thinking necessary in solving equations? Can all equations be solved in exactly the same way?Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 9: Literal Equations and Rearranging FormulasTeachers GuideOpening Activity: Solve each equation for ?. For part (c), remember a variable symbol, like ?, ?, and ?, represents a number. a. ????=?? b. ?????=??? c. ????=? Decision Maker: These have been placed side by side so the similarities in the work can be more easily discussed. This may not work for students who have big writing so you’ll need to decide if this will work for your students (in which case you might want to format the page to landscape). Coach: Since we are building the relationship between arithmetic ways of working and algebraic ways of working we want students to reinforce that the way you approach solving an equation with numbers is the same as you do with variables. When students get stuck focus your questions back to working with numbers and connecting that to working with an equation with a variable.Decision Maker: Could 2x-6=10, 2x-b=10, and 2x-b=c be better for your students? What do you gain by using these equations? What do you lose by using these equations? Which sequence of equations will help your students better understand that what they do with numbers can be done with letters?What similarities do you see in your work in solving each of the equations?Activity 2:Solve for P in the simple interest formula: A = P(1 + rt)Challenge: Try to isolate the t.Solve for x 3x + 4 = 6 – 5x Solve for xax + b = d – cx Coach: Note how we also place these two problems side-by-side. The big step in this one will occur with the need to factor out the x in ax + cx so the student will be able to isolate the x. The way students can understand this is when he/she solves the first equation for x and you combine 3x and 5x it is the same as writing x(3 + 5), which is what must be done in the literal equation. This might help students see the connections.4590904444500154808142990026813126496We factor out the x, the common factor 00We factor out the x, the common factor Activity 3:Michael and Alicia are working together and are complaining about rearranging formulas.Their teacher wrote the following on the board:8792340591y = mx + b b = ymx x = y+bmAre all the statements true? Justify your answer. If any of them is wrong, correct it and explain what you did.00y = mx + b b = ymx x = y+bmAre all the statements true? Justify your answer. If any of them is wrong, correct it and explain what you did.Michael said to Alicia, “We better find someone to help us” They come over to you. Help them make sense of this problem. Note to Teacher: Flexibility is especially important when it helps you arrange things in a more useful way.What is y = mx + b? What is its meaning? Why might it be important to be able to rearrange this formula? Is it possible to solve for m in the equation? If so, what would be its value? Journal Writing/Exit Ticket: You know that the perimeter of a rectangle is P = 2L + 2W. How would you explain to someone how to rearrange the formula if you wanted to work from the width alone? Homework:Regents Practice:Note to Teacher: Here are several items that have appeared on previous CC Algebra I Regents exams. Though they aren’t open-ended problems, it is important for students to be exposed to the format and the language of the questions. We want students to learn something from the experience, not just answering the questions.Decision Maker: How do you want to facilitate the practice experience? Perhaps a discussion on the phrasing of the questions (presentation of equation followed by instructions), certain phrases that are used: “in terms of”, “may be expressed as”.42146242928800left129191500left10124500left508000Unit 2: Equality, Inequality & Problem SolvingLesson 9: Literal Equations and Rearranging FormulasStudent VersionName_______________________Date________________________Opening Activity: Solve each equation for ?. For part (c), remember a variable symbol, like ?, ?, and ?, represents a number. a. ????=?? b. ?????=??? c. ????=? What similarities do you see in your work in solving each of the equations?Activity 2:Solve for P in the simple interest formula: A = P(1 + rt)Now here is your challenge try to isolate the t.Solve for x.3x + 4 = 6 – 5x Solve for x.ax + b = d – cxActivity 3:Michael and Alicia are working together and are complaining about rearranging formulas.Their teacher wrote the following on the board:8763064770y = mx + b b = ymx x = y+bmAre all the statements true? Justify your answer. If any of them is wrong correct it and explain what you did?00y = mx + b b = ymx x = y+bmAre all the statements true? Justify your answer. If any of them is wrong correct it and explain what you did?Michael said to Alicia, “We better find someone to help us” They come over to you. Help them make sense of this problem. What is y = mx + b? What is its meaning? Why might it be important to be able to rearrange this formula? Is it possible to solve for m in the equation? If so, what would be its value?Journal Writing/Exit Ticket: You know that the perimeter of a rectangle is P = 2L + 2W. How would you explain to someone how to rearrange the formula if you wanted to work from the width alone? Homework: Regents Practice:left471490500left230200700left1012450031897746275300Transitioning to Inequalities: Note to TeacherIn the next few lessons you and your students are going to be working with inequalities leading into solving problems with inequalities at the end of this unit. A question to reflect on is: how does solution evolve as we move from equations to inequalities? As one moves from linear equations to linear inequalities one observes a very interesting evolution in what a solution is. An example will help illustrate this:Starting with 3x + 4 = 16, and solving for x leads to x = 4. Graphing x = 4 we need a number line that represents the possible values of x, but there’s only one value or place that makes the equation true. Thus, what makes the equation true is one point on the number line.467833883831-10234x001-10234xIf we change 3x + 4 = 16 to 3x + 4 < 16, and then solve for x the solution changes to x < 4. Here again when we go to graph this we need a number line representing the possible values of x, but this time there are many values or a region of the number line that makes the inequality true. Thus, what makes the inequality true is a region of the number line.Further considerations for where the next section goes:What changes when we transition to thinking about 3x + 4 = y and 3x + 4 > y? What connections are there from working with 1 variable (1-dimensional space – the number line) to working with 2 variables (2-dimensional space – the coordinate plane)? How do our notions of solution evolve as we move from 1-D space to 2-D space?Then, what changes when we transition to systems? For instance: 3x+4=y-5x=y and 3x+4>y-5x>y. How do our notions of solution change when another equation or inequality must to be true simultaneously? How are these new notions of solution similar to our previous thoughts on solution? How are the solutions here analogous to when we worked with 3x + 4 = 16 and 3x + 4 > 16?In thinking about your students:How can tracing the evolution of solution as you work through these next few lessons and unit help create coherency for your students? How can these observations of solution help your students create expectations of the solution to the problems they will undertake?Decision Maker: You will need to make a choice based on the experience you’ve had with your students thus far on what should come next. What you decide should set your students up for what’s coming in Lessons 11, 12, 13 & 14. It would be a good idea to look ahead at Lessons 11, 12, 13 & 14 and use that what’s coming to assist you in this decision.If you think that your students only need a brief review of inequalities then use the review lesson that follows. If you think that your students need a more substantive experience with inequalities use lessons A & B to get your students ready for the rest of the lessons on inequalities. Lessons A & B are in the appendix which follows Lesson 21 at the end of this unitTo help make this decision you may want to add a quick exit ticket to the previous lesson that looks something like:What does each of the following mathematical symbols mean?< < > >=How would you represent the following using mathematical symbols?15098231340151-10234z001-10234zHow would you represent the following mathematically?You must be more than 4 ft. tall to go on a ride at Great Adventures.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 10: Review of Inequalities LessonTeachers GuideOpening Activity:What do the ideas “equality” and “inequality” mean to you?How do you see these ideas outside of mathematics? How do you see these ideas within mathematics?Provocateur: The purpose here is for students to think about these ideas first in a global way then within the context of mathematics. These ideas are inherent to the world students live in so you have a chance to hear how they think about them. This activity is also a set up for the next activity in which students are asked to group different statements. Activity 2:Note to Teacher: The overarching goal of Activity 2 is to better understand how developed student thinking is with respect to inequalities. Use the discussion after Part 2 to work through the misconceptions in the room so that by the end of the activity there’s a basic flexibility in thinking about inequalities.Decision Maker: Groups of students will work on Parts 1 and 2 of this activity before coming back together to discuss their work. How will students be grouped? Based on the work you see, which students will you ask to share out? What are you looking for from the groups prior to transitioning to a whole group discussion?Learner: Parts 1 and 2 of this activity will give you a clearer sense of where your students are in their understanding and representations of equalities and inequalities. Stand back and let them make sense of each of these statements and how to represent them. Be sure to note the misconceptions you are seeing and try to set it up so that another group or student would be able to address the misconception in the upcoming group discussion. As best you can, try to seed the upcoming conversation so that as much as possible students are addressing other students’ misunderstandings and/or questionsThe 12 statements below include a couple of equality statements and many inequalities. We’ve included the mathematical statement that was intended to be communicated. Feel free to adjust the language if you think the wording is not clear.Read at the different statements below. Decision Maker: How do you want your students to interact with these 12 statements? Do you want them to be able to move them around (for tactile learners - they’ll each need to be on slips of paper), or do you want them to work with them in the list form in which they are provided?You need to be more than 5 ft. tall to go on the ride at Great Adventures. (height > 5 ft)The elevator can hold up to 1500 pounds. (total pounds in elevator ≤ 1500 :: You’ll need to decided what you want to do with the implications of negative weight. Does the context actually turn this into a compound inequality? So 0 < total pounds in elevator ≤ 1500?)Stephanie estimated the cost of the book with tax to be greater than $10.50 but less than $12.00. ($10.50 < price of book < $12.00)John earned $40,000 last year. ( John’s income = $40,000)The road trip from NYC to Los Angeles will take at least 7 days but no more than 15 days. ( 7 days ≤ road trip from NYC to Los Angeles < 15 days)The new contract for the basketball player was rumored to earn him more than $5 million, and up to $8 million per season. ( $5 million < $ per season ≤ $8 million)The football game was viewed by 60,000 or more people. (football viewers ≥ 60,000)The temperature is not going to be higher than 30° F tomorrow. (temperature tomorrow < 30° F)Lilliana falls in the group of people who earn greater than $50,000 and less than $100,000. ($50,000 < Liliana’s income < $100,000)Miguel is 5’5” tall and Jamal is 5’8” tall. (There are several ways to represent this depending on how the student interrupts it. Among the interruptions the students may have… Miguel’s height = 5’5” and Jamal’s height = 5’8”, or Jamal’s height > than Miguel’s height [how would this be graphed?] )Jorge estimated he was 15 to 20 minutes late. (15 mins ≤ mins Jorge estimated he was late ≤ 20 mins)The GPS device indicated that the upcoming traffic jam was going take more than 30 minutes to get through. ( time through traffic jam > 30 minutes)Part 1: Group the ones you think can be mathematically represented as a statement of equality together and the ones that can be mathematically represented as an inequality together. Be prepared to justify why you placed each statement in the group you chose, especially those you put in both groups.Part 2: Create a mathematical representation for each of the statements based on the group you placed it.Learner: It will be interesting to see where the students take this. Do they use the symbols for equality and inequality? Does anyone create a graph on a number line? Do they create another kind of visual representation? All of these observations will help you better understand how your students are thinking about equality and inequality. Additionally, take note of which groups start with symbols, which groups start with a graph, or other visual representation.Provocateur: If a group uses symbols, ask them to create a corresponding graph. If a group creates a graph, ask them to write a mathematical statement using symbols. This will help give you a clearer picture of your students’ thinking on this. If a group creates a picture of the statement, ask them how they could translate their picture into a mathematical representation. Decision Maker: Students’ misunderstandings create rich opportunities for discussion and clarification. What opportunities do you see in the students’ work thus far? What issues are unique to some groups? What issues are common to all of the groups? How will each of these types of issues presented be handled in the discussion following Part 2?Decision Maker: You may want to include these real-world examples for your students.Part 3: Create a mathematical representation for each of the statements.-6352000250037052261479550025050751079500Decision Maker: After the discussion and working through the misconceptions the students have about inequalities you may want to use this Part 4 to check how they made sense of the conversation.Part 4: Create two “real world” inequality statements and the mathematical representation of each statement. Final Activity: Performance Task- Create a ScenarioCreate a scenario and the line graph for each of these inequalities:Decision Maker: If there is time you should have a few students share the situations they created for each of these. x < 10011 < x < 190 < x < 1000Learner: The accuracy with which your students represent these inequalities will further help you understand what sense the students have made of inequalities. Keep in mind the lingering misconceptions you see as you prepare for the next lesson. What experiences do you think students need to help them work through their misconceptions? (In the first example, the solution set includes all real negative numbers to negative infinity.) How can another student’s work help in this process?Journal writing: Examine the two statements below:Caleb has at least $100.Tarek has more than $100.Explain the difference between these two statements.Express each of the two statements with a graph and mathematically using numbers and symbols.Decision Maker: Below is a possible homework. What do you want for your students in this homework experience? What do you want to learn about your students by looking at their work? Does this assignment accomplish those goals? What would need to change about this assignment in order to better meet your goals for this homework experience?Homework: Part 1: How would you represent each of the following number line graphs symbolically? What similarities and differences do you see between the following 4 number lines?Part 2: How would you represent each of the following number line graphs symbolically?What similarities and differences do you see between the following 4 number lines?Unit 2: Equality, Inequality & Problem SolvingReviewing Inequalities Student VersionName_______________________Date________________________Opening Activity:What do the ideas of “equality” and “inequality” mean to you?How do you see these ideas outside of mathematics? How do you see these ideas within mathematics?Activity 2:Read at the different statements below. You must be more than 5 ft. tall to go on the ride at Great Adventures.The elevator can hold up to 1500 pounds.Stephanie estimated the cost of the book with tax to be greater than $10.50 but less than $12.00.John earned $40,000 last year.The road trip from NYC to Los Angeles will take at least 7 days but no more than 15 days.The new contract for the basketball player was rumored to earn him more than $5 million, and up to $8 million per season.The football game was viewed by 60,000 or more people.The temperature is not going to be higher than 30° F tomorrow.Lilliana falls in the group of people who earn greater than $50,000 and less than $100,000.Miguel is 5’5” tall and Jamal is 5’8” tall.Jorge estimated he was 15 to 20 minutes late.The GPS device indicated that the upcoming traffic jam was going to take more than 30 min. to get through.Part 1: Group the ones you think can be mathematically represented as a statement of equality together and the ones that can be mathematically represented as an inequality together. Be prepared to justify why you placed each statement in the group you chose, especially those you put in both groups.Part 2: Create a mathematical representation for each of the statements based on the group in which you placed it.Part 3: Create a mathematical representation for each of the statements.-6352000250037052261479550025050751079500Part 4: Create two “real world” inequality statements and the mathematical representation of each statement. Final Activity: Performance Task- Create a ScenarioCreate a scenario and the line graph for each of these inequalities:x < 10011 < x < 190 < x < 1000Journal writing: Examine the two statements below:Caleb has at least $100.Tarek has more than $100.Explain the difference between these two statements.Express each of the two statements with a graph and mathematically using numbers and symbols.HomeworkPart 1: How would you represent each of the following number line graphs symbolically?89611818542000What similarities and differences do you see between the following 4 number lines?Part 2: How would you represent each of the following number line graphs symbolically?What similarities and differences do you see between the following 4 number lines?825338000Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 11: AND & OR Teacher GuideRepresenting sets algebraically and graphically5057775203835Opening Activity: Read the following story then answer the questions Jose, Latoya, Robert and Mai are having a conversation about numbers.Jose: I am thinking of all the numbers that are greater than 3 and less than 20.Latoya: I am thinking of all the numbers that are less than 3 and greater than 20.Robert: I am thinking of all the numbers that are in both of your groupsMai: I am thinking of all the numbers that are in either Jose’s group or Latoya’s group.It is extremely important for students to know that, unless otherwise stated, when discussing the solution set to an inequality we refer to the set of real numbers. For example, Jose could be thinking of 3, 3.1, 3.004, etc.What are the numbers in Robert’s mind? What are the numbers in Mai’s mind? Learner: What sense are your students making of inequalities?Decision Maker: The groups should share their findings with reasoning and a discussion should arise about the differences between the ideas of Robert and Mai. How would you represent your solutions mathematically? The solutions would probably be best represented in sets with Robert’s solution being an empty set.Coach: We want students to be able to represent solutions to inequalities in multiple ways. For example:-3<x<2 can also be written as (-3,2] or it can be written as x>-3 AND x< 2. If a student has a solution like x>12 you might want to show them that it can also be written as (12,∞). You might need to have a discussion about notation and have a conversation about it when necessary.Activity 2: What is the difference between AND and OR?The teacher has to determine which of two students, A or B, will be win the candy bar. She placed 10 pieces of paper in a bag, each paper numbered 1-10. The two students will pick two numbers. Who has a better chance of winning?Person A: To win the game he or she must pick a 2 or a 6.Person B: To win the game he or she must pick a 2 and a 6.What can you now say is the difference between “and” and “or”?Activity 3:Given the following information, graph the solution to this problem. Explain what you did and why you did it. x >3 /\ x < 8 (Note: /\ means and)Learner: It is important to see how students take a new type of question in which they have the knowledge to answer it yet have not seen the question in this form.Could you come up with another way of writing the given statement algebraically using just one x? Explain why you think your answer is correct.Activity 4:We are going to look at a similar problem from the one you worked on previously except we are going to change one thing. Try to represent graphically and be ready to discuss how the problem was different and the effect it had on the solution.x >3 \/ x < 8 (Note: \/ means or)Activity 5:Look at the following problem. How would you graph it? x < 3 \/ x > 8 Final ActivityNow come up with the graphical representation of the solution for this problem. -2 > x /\ 18 < xLearner: This task has no solution. Let us see what the students do with this.Closing Activity:Journal Writing:Look at the different tasks you experienced today. Write in your own words the effect of AND and OR on an inequality statement.Unit 2: Equality, Inequality & Problem SolvingLesson 11: AND & ORStudent VersionName_______________________Date________________________5181600116205Opening Activity: Read the following story then answer the questionsStory: Jose, Latoya, Robert and Mai are having a conversation about numbers.Jose: I am thinking of all the numbers that are greater than 3 and less than 20.Latoya: I am thinking of all the numbers that are less than 3 and greater than 20.Robert: I am thinking of all the numbers that are in both of your groupsMai: I am thinking of all the numbers that are in either Jose’s group or Latoya’s group.What are the numbers in Robert’s mind? What are the numbers in Mai’s mind? Justify your answer. Activity 2: What is the difference between AND and OR?The teacher has to choose who would be given the candy bar. She placed 10 pieces of paper in a bag, each paper numbered 1-10. The two students will pick two numbers. Who has a better chance of winning?Person A: To win the game he or she must pick a 2 or a 6.Person B: To win the game he or she must pick a 2 and a 6.What can you now say is the difference between “and” and “or”?Activity 3:Given the following information, graph the solution to this problem. Explain what you did and why you did it. x >3 /\ x < 8 (Note: /\ means and) Could you come up with another way of writing the given statement algebraically using just one x? Explain why you think your answer is correct.Activity 4:We are going to look at a similar problem from the one you worked on previously except we are going to change one thing. Try to represent graphically and be ready to discuss how the problem was different and the effect it had on the solution.x >3 \/ x < 8 (Note: \/ means or)Activity 5:Look at the following problem. How would you graph it?x < 3 \/ x > 8 Final ActivityNow come up with the graphical representation of the solution for this problem.-2 > x /\ 18 < x Closing Activity:Journal Writing:Look at the different tasks you experienced today. Write in your own words the effect of AND and OR on an inequality statement.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 12: How does the domain change our thinking and affect the solution sets?Teacher GuideOpening Activity:Look at the two following questions and explain how they are different including their solutions:How many whole numbers are there between 3 and 4?How many numbers are there between 3 and 4?Decisions Maker: This can lead to a discussion about how solution sets differ based on a given constraint. Here you can define the notion of a domain as a given constraint. When we move into functions we will expand the definition. The domains in inequalities affect the solution sets. In activity one we see that in the first example there are an infinite amount of real numbers between any two real numbers but there are no whole numbers between 3 and 4. This should be part of the discussion about the domain affecting the solution set.Activity 2:With your partner look at the next two problems, find the solution set and graph the result. Be ready to explain your reasoning for each solutionAll numbers greater than 4 and less than 7.All whole numbers greater than 4 and less than 7.Decision Maker: Have students share their ideas. It is important to note that the work we already did up to this point referred to all (real) numbers so they know that already. Will they be able to differentiate between the two problems, how they write the solution and how they graph the solution?Activity 3: Look at the following solution sets and describe the difference. Describe what the domains could be for each of these statements. 3 < x < 19 3, 4, 5, 6,…, 19 Activity 4:For the inequality -3 < x < 8Graph the solution if the domain is:All whole numbers {0, 1, 2, 3, …}All numbers on the number line {All real numbers)All positive integers less than 5All negative numbersAll integers less than 8Activity 5:Look at the following graphs of solution sets. Jose said that the set of all integers could be the domain for both graphs while Natasha felt that there is no domain that could have worked for both. What do you think? 818707242771-10234x001-10234x 8080741306631-10234x001-10234xClosing Activity:For each of the following statements:State the possible domains and solution sets for each.Graph the solution sets.You need to be at least 5 ft. tall to go on the ride at Great Adventures.The elevator can hold up to 1500 pounds. He has at most 10 tickets to give away. (the domain would be counting numbers less than or equal to 10, or counting numbers less than 11)It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.The store has at least 50 different bicycles to choose from but no more than 75 (the domain would be counting numbers/integers greater than or equal to 50 and less than or equal to 75)Lilliana falls into the group of people who earn between $50,000 and $100,000.Note to teacher: It is important to have a discussion about how solution sets differ based on a given context. Why are the domain of Allen and Carmen’s situations different?Exit Slip: Is the domain the same for each of the following statements? Explain.Allen needs at most 5 pounds of apples for the apple cobbler recipe. Carmen needs at most 5 forks to share the apple cobbler with her friends.Unit 2: Equality, Inequality & Problem SolvingLesson 12: How does the domain change our thinking and affect the solution sets?Student VersionName_______________________Date________________________Opening Activity:Look at the two following questions and explain how they are different including their solutions:How many whole numbers are there between 3 and 4?How many numbers are there between 3 and 4?Activity 2:With your partner look at the next two problems, find the solution set and graph the result. Be ready to explain your reasoning for each solutionAll numbers greater than 4 and less than 7All whole numbers greater than 4 and less than 7.Activity 3: Look at the following solution sets and describe the difference. Describe what the domains could be for each of these statements. 3 < x < 19 3,4,5,6,…19 Activity 4:For the inequality -3 < x < 8Graph the solution if the domain is:all whole numbers {0, 1, 2, 3, …}All numbers on the number line {All real numbers)All positive integers less than 5All negative numbersAll integers less than 8Activity 5:Look at the following graphs of solution sets. Jose said that the set of all integers could be the domain for both graphs while Natasha felt that there is no domain that could have worked for both. What do you think?7198201311581-10234x001-10234x776514232231-10234x001-10234xClosing Activity:For each of the following statements:State the possible domains and solution sets for each.Graph the solution sets.1) You need to be at least 5 ft. tall to go on the ride at Great Adventures.2) The elevator can hold up to 1500 pounds.3) He has at most 10 tickets to give away. 4) It will take us at least 10 days and no more than 12 days to drive from NY to Los Angeles.5) The store has at least 50 different bicycles to choose from but no more than 75.6) Lilliana falls into the group of people who earn between $50,000 and $100,000.Exit Slip: Is the domain the same for each of the following statements? Explain.Allen needs at most 5 pounds of apples for the apple cobbler recipe. Carmen needs at most 5 forks to share the apple cobbler with her friends.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 13: Can you discover how to solve inequalities? Teacher GuideOpening Activity:Look at this statement. What does it mean? x + 3 > 8What are possible solutions to this problem? How did you get them?How many solutions does it have? How do you know?How does x + 3 > 8 compare with x + 3 = 8? How are they similar and how are they different?How can you use your knowledge of equations to solve the inequality?Decision Maker: A good discussion should arise from these questions. You want students to think about how equations and inequalities are similar and different.How would the solution sets differ if the domain was all whole numbers versus if the domain was all real numbers?Activity 2Look at these two statements. You know how to solve the equation, now see if you can figure out how to solve the inequality. 5x + 4 = 24 5x + 4 < 24 What is the solution for each of them? How are the solutions similar and how are they different?How can we be sure that the solution(s) are correct?Questioner: Here it is important to help students to check their answers to inequalities to make sure that their answer makes sense. This will become very important when we multiply or divide by a negative.If you were asked to compare how you solved an equation and how you solved an inequality what would you say? How are they similar and how are they different?Activity 3: An OutlierSolve the following inequality, find your solution and check if your answers make sense.-3x < 12Questioner: When students find that their solution did not work, it should lead to an interesting discussion. Why didn’t it work? What would be the correct solution? Why does this inequality act differently from the other inequalities? Can you come up with a possible conjecture about certain inequalities? Can you test out your ideas on another problem and see if your idea works? Discuss the different results.Activity 4:Now in your group compare these two inequalities. Can you explain how the solutions differed and why that happened? -4x > -12 4x > -12Activity 5:In your group attempt these two problems and be ready to discuss the solutions x > 12 x > 12 3 -3Activity 6: PracticeSolve each of these inequalities. Show your work and explain how you know your solution set is correct. Assume the domain is all real numbers unless another domain is specified.3x – 6 > 12-5x – 3 < 2x – 62x-3 > 94 – 2x > 3x – 6 (Domain is all whole numbers)Closing Activity/Journal Writing: You have been asked to teach someone how to solve inequalities and why your approach works. Describe what you would tell this person.Unit 2: Equality, Inequality & Problem SolvingLesson 13: Can you discover how to solve inequalities?Student VersionName_______________________Date________________________Opening Activity:Look at this statement. What does it mean? x + 3 > 8What are possible solutions to this problem? How did you get them?How many solutions does it have? How do you know?How does x + 3 > 8 compare with x + 3 = 8? How are they similar and how are they different?How can you use your knowledge of equations to solve the inequality?How would the solution sets differ if the domain was all whole numbers versus if the domain was all real numbers?Activity 2:Look at these two statements. You know how to solve the equation, now see if you can figure out how to solve the inequality. 5x + 4 = 24 5x + 4 < 24 What is the solution for each of them? How are the solutions similar and how are they different?How can we be sure that the solution(s) are correct? If you were asked to compare how you solved an equation and how you solved an inequality what would you say? How are they similar and how are they different?Activity 3: An OutlierSolve the following inequality, find your solution and check if your answers make sense?-3x < 12Activity 4:Now in your group compare these two inequalities. Can you explain how the solutions differed and why that happened? -4x > -12 4x > -12Activity 5:In your group attempt these two problems and be ready to discuss the solutions x > 12 x > 12 3 -3Activity 6: PracticeSolve each of these inequalities. Show your work and explain how you know your solution set is correct. Assume the domain is all real numbers unless another domain is specified.3x – 6 > 12-5x – 3 < 2x – 62x-3 > 94 – 2x > 3x – 6 (Domain is all whole numbers)Closing Activity/Journal Writing: You have been asked to teach someone how to solve inequalities and why your approach works. Describe what you would tell this person.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lessons 14: Solving Compound InequalitiesTeachers GuideOpening Activity:Mai & Latoya were playing around with the different inequalities. They noticed that they’ve solved inequalities like 5 < 2x + 1 as well as 2x + 1 < 9, but aren’t sure what would happen if they put them together like 5 < 2x + 1 < 9.How would you solve for x in 5 < 2x + 1 < 9? What would you tell Mai & Latoya?Leaner: What do the students do with this? Do any of them break it apart? Do any of them connect it back to working with AND?Coach: Students may be frustrated with the inequality formulated in this way. So you may ask, how can you make this inequality simpler? This can be a difficult question for students since they often don’t feel like they have the agency to change the problem, but if they are going to leverage the power of simplifying the problem it’s important that they feel like they have that power. Here are a couple of suggestions for how this could be simplified to get you thinking about how you can coach your students:Would breaking them apart help us?What would you do if we simplified the inequality to 9 < 3x < 12? If they are able to make sense of that one, you could follow up with 10 < 3x + 1 < 13. How is this one different than the last one? What clue does that give us for where we could start thinking about solving this one? If they are able to make those connections ask them how they could use this thinking on Mai & Latoya’s question.Coach: We want students to be able to represent solutions to inequalities in multiple ways. For example:-3<x<2 can also be written as (-3,2] or it can be written as x>-3 AND x< 2. If a student has a solution like x>12 you might want to show them that it can also be written as (12,∞).Decision Maker: When the time seems right you’ll want to bring the students back together to talk about the different strategies and connections the students have made. Coming out of this Opening Activity you want students to feel the freedom to break apart the inequality as well as connecting this solving process to the work they did with AND in Lesson 11. Once the connection to AND is made graphing the solution can be easier. Decision Maker: How do you want to memorialize these conclusions that students make as a part of the discussion about the Opening Activity?Provocateur: An interesting question you may want to pose to your students after the discussion about the opening activity:In a previous lesson we’ve see that 2 < x < 4 when graphed looks like the graph below:10738881881081-10234x001-10234xHow do you make sense of the connection between 5 < 2x + 1 < 9 and 2 < x < 4?Activity 2:Make a conjecture about what you think would happen if Mai & Latoya’s inequality changed to 5 < -2x + 1 < 9.Note to Teacher: One of the cool things for our students to see as they grapple with negatives is how it influences solutions. When the coefficient changes sign it acts as a reflection about 0. INTERESTING! So if we know the solution to one equation/inequality reflecting it about zero will provide us with the solution when the coefficient changes sign. Students should also represent the solution algebraically.5 < 2x + 1 < 966985150651-10234x001-10234x5 < -2x + 1 < 9723014149181-2-4-3-101x00-2-4-3-101xTest out your conjecture by solving Mai & Latoya’s inequality. Represent your solution both graphically and algebraically.5 < -2x + 1 < 9Compare the graph from the Opening Activity to the graph from your solution here. What does this change tell us about how we can think of the effect of negatives? How does this make sense with what you’ve already learned about solving inequalities with negative coefficients?Activity 3:Based on what we learned from the issue Mai & Latoya raised, how would you handle x – 1 > -1 AND x + 6 < 8? How would you represent the solution graphically and algebraically?Decision Maker: Here again when you feel the time is right have the students share their thinking.Activity 4:How would you solve with x – 2 > 6 OR 2x < 6? What’s the graph of this inequality look like? How would you represent the solution graphically? Algebraically?Activity 5:Decision Maker: You’ll need to gauge what level of practice and how many exercises your students will need. You can fill those in here for Activity 4 or include it in homework. Either way make time for students to see each other’s work and talk about their processes.Closing Activity/Journal writing: How do AND and OR change the work we do with inequalities? Unit 2: Equality, Inequality & Problem SolvingLesson 14: Solving Compound InequalitiesStudent VersionName_______________________Date________________________Opening Activity:Mai & Latoya were playing around with the different inequalities. They noticed that they’ve solved inequalities like 5 < 2x + 1 as well as 2x + 1 < 9, but aren’t sure what would happen if they put them together like 5 < 2x + 1 < 9.How would you solve for x in 5 < 2x + 1 < 9? What would you tell Mai & Latoya?Activity 2:Make a conjecture about what you think would happen if Mai & Latoya’s inequality changed to 5 < -2x + 1 < 9.Test out your conjecture by solving Mai & Latoya’s inequality. Represent your solution both graphically and algebraically.5 < -2x + 1 < 9Compare the graph from the Opening Activity to the graph from your solution here. What does this change tell us about how we can think of the effect of negatives? How does this make sense with what you’ve already learned about solving inequalities with negative coefficients?Activity 3:Based on what we learned from the issue Mai & Latoya raised, how would you handle x – 1 > -1 AND x + 6 < 8? How would you represent the solution graphically and algebraically? Activity 4:How would you solve with x – 2 > 6 OR 2x < 6? What’s the graph of this inequality look like? How would you represent the solution graphically? Algebraically?Activity 5: Practice Closing Activity/Journal writing: How do AND and OR change the work we do with inequalities? Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lessons 15 – 19: Algebraic Problem SolvingNote to Teacher: We’ve allotted 4 lessons to work on algebraic problem solving. You’ll need to decide if this amount of time is too much, too little, or just right for you and your students. There are two goals for these lessons:to help students think about what the variable is in the problem. to help students use tables to expose the arithmetic pattern in the problem that they are working to generalize. The next unit will go much further in working with tables and linear relationships, but for now these two goals are the foci.Reflector: It’s important here to recognize what we’re trying to accomplish when we say algebraic problem solving. Most students as they enter high school are primarily arithmetic thinkers. By this we mean that they are more comfortable working with specific numbers than working with variables to generalize a group of numbers. Thus, one of the major goals of high school algebra is to help student thinking to grow to a place where their thinking can primarily be described as algebraic rather than arithmetic. This transition from arithmetic to algebraic thinking is why using a table becomes such a powerful tool. The table can capture the arithmetic elements of a problem in a way that allows the pattern embedded in the elements to become apparent. This pattern is then what gets generalized using algebraic symbols, which then can be manipulated to find out anything about the relationship described in the problem. Here’s a visual that may be helpful in better understanding this idea:right106983SpecificArithmeticGeneralAlgebraicA table is used to capture and organize the arithmetic information given in a problemThe pattern is translated into algebraic symbolsPatternsFinding and understanding the pattern in the table moves one to a place where they are ready to generalize00SpecificArithmeticGeneralAlgebraicA table is used to capture and organize the arithmetic information given in a problemThe pattern is translated into algebraic symbolsPatternsFinding and understanding the pattern in the table moves one to a place where they are ready to generalizeHere are few questions that would be worth spending some time thinking about:How is the definition of mathematics as, “the science of patterns”, central to developing student understanding of algebra?How is the definition of algebra as, “generalized arithmetic”, central to developing student understanding of algebra?Decision Maker: We’ve provided 9 problems below that are a bit more straight- forward than the types of problems we usually have students work on. You’ll need to decide which problems you want to use and you may need to supplement these problems. Additionally, it’s strongly encouraged that you continue to use the KNN chart with your students. To save space we’ve included a blank table here that you can copy and paste into the handouts you make from the problems below.KnowNeed to KnowNext StepsAs often as it seems effective it will be useful to ask the students or groups of students share their work so they can influence one another’s problem solving processes. What’s more powerful for students is seeing the work of other students, rather than seeing an adult’s process.Problem 1: Jose and Nadia’s DVDsJose has twice as many DVDs as Nadia. If they have 39 DVDs together, how many DVDs does each have?Learner: Let students play with this. They can use guess and check or any other method they want Have students present their thinking, you want to see where your students go with this.Decision Maker: If a table arose out of the student presentations, use it to get to an algebraic representation of the underlying relationship. If a table was not a part of the students work and how they thought through the problem then you’ll want to bring that into the discussion.Here’s a possible approach to bringing the table into the discussion:Let’s make a table to represent possible values for the DVDs of Nadia and Jose. We can pick any values for Nadia to try to find the answer. If I say, Nadia has 6 DVDs, how many would Jose have? Does that give us 39? Now fill in the table using any values for Nadia and calculate the corresponding value for Jose. NadiaJoseTotal61218Suppose the value is: xWhat goes here?Use these variables to make an equation that will help us solve the problem.Solve your equation. Compare it with the answer you got earlier.Problem 2: A Problem of PenniesModeler of Mathematical Thanking: Many students struggle with not only reading these types of problems, but with making sense of an approach to getting started with the thinking necessary to enter into the problem. We can model supposition as an approach to this problem:If Rafael had 1 penny, how many will Susan have? How many will Tashawn have?How many would they have in total?Assume Rafael has 1 penny.This means Sonia has __ pennies.Tashawn has __ pennies.Assume Rafael has x pennies.This means Sonia has __ pennies.Tashawn has __ pennies.For students who struggle with making sense of the language or structure of the question, it sometimes becomes hard to determine if the struggle is based on the mathematics or the reading (or other) skills involved in entering into the problem. You can teach students how to break it down into smaller, simpler parts. The scaffold shown below, in which the text is chunked and students have to pull text from the question, is helpful for ELLs (red font is what we want students to be able to pull from the text):Rafael has some pennies.Sonia has 4 more pennies than Rafael.Tashawn has three times as many pennies as Rafael.Sonia has 4 more pennies than Rafael and Tashawn has three times as many pennies as Rafael. All together they have 34 pennies. How many does each person have?Problem 3: Cows and ChickensThere are 50 cows and chickens on the farm. Jose forgets how many of each animal he has but he was told that the total number of legs of all the animals is 148. How many cows and how many chickens are on the farm?Problem 4: Pia’s Age16 years from now, Pia’s age will be twice her age 12 years ago. Find her present age. Problem 5: How many years old?Jon is 27 years older than Susan. In five years Jack will be 4 times as old as Susan. How old is Jon? How old is Susan?Problem 6: How old is her daughter?Five years from now, the sum of the ages of a woman and her daughter will be 40 years. The difference in their present age is 24 years. How old is her daughter now?Problem 7: Consecutive OddsFind three consecutive odd integers whose sum is 71.Problem 8: A Problem with IntegersFind three consecutive even integers in which twenty more than twice the sum of third is 28 more than the sum of the first two integers.Problem 9: Donna’s MixDonna wants to make trail mix made up of almonds, walnuts and raisins. She wants to mix one part almonds, two parts walnuts, and three parts raisins. Almonds cost $12 per pound, walnuts cost $9 per pound, and raisins cost $5 per pound. Donna has $15 to spend on the trail mix. How much does Donna’s trail mix weigh?Problem 10: A Fractional Life (from IMP – Year 1)Note to Teacher: Students might find this weird because they are setting up an equation equal to x. It is okay, and natural, for students to assign a unique variable to each unknown (b = boy, y = youth, m = man, a = age). However, they will have an equation in four variables. Can they assign an expression for each unknown?Here is a problem that is part of the Greek Anthology, a group of problems collected by ancient Greek mathematicians.Demochares has lived a fourth of his life as a boy, a fifth as a youth, a third as a man, and has spent 13 years in old age. How old is Demochares?Decision Maker: Maria’s Dilemma is much more open-ended than the previous problems and in fact there isn’t just one solution to the problem. This problem could be used as an on-demand performance task done individually or in groups. Be sure to discuss this task with your Math coach.Problem 11: Maria’s DilemmaMaria was collecting money for the homeless and she lost the can with the money. She is frantic. She is trying to figure out as best she can how much money she had. She doesn’t want to cheat anyone but she doesn’t have much money to spare. She knows she had nickels, quarters and dimes in the can the last time her friend checked an hour ago. She had five more dimes than quarters and an equal amount of quarters and nickels. Her friend also checked the total amount and saw that there was $2.50. Other people have stopped by and put in the same amount of quarters and nickels that there was when her friend checked earlier. Three people gave dimes. She estimates that at most there was $4.00 in the can. Is she correct? Can you help Maria and suggest how much she should give to the homeless based on the information she had?Coach: Can you just use one variable? Learner: How are students assigning variables to the problem? Ask them if there is a coin (variable) that can serve as a reference point for all the coins (the quarter – both the number of dimes and the number of nickels are directly linked on the number of quarters).Decision Maker: An interesting discussion about how students approached this problem and what conclusions they made could follow their work on this problem.Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 20: Inequality ProblemsTeacher GuideLearner: Here are three problems. They all are inequalities with multiple solutions. How will students approach these problems? They could use tables or they can attempt algebraic approaches. Let them work on each problem alone at first. Then after five minutes let them join a partner. There should be a class discussion for each problem. Opening Activity:You want to join a health club to get back into shape.The Tone Up Club charges $50 to join plus $9 per month.Be in Shape has no sign up fee and charges $12 per month.You have a maximum of $200 to spend. Which club will you join? Why?Activity 2: A) You gave your brother a 50-foot start in a race. If he ran 10 feet every 5 seconds what is the minimum speed you would have to run to catch him in at least 20 seconds from the time you both started?B) How long would it take you to catch your brother if you ran as fast as Usain Bolt. Usain runs 300 feet in less than 10 seconds.Activity 3:Given 2x + ax – 7 > -12, determine the largest integer value of a when x = -1. Decision Maker: Here are a few items that have appeared on the CC Algebra I exams. How do you want to use them? Perhaps you might consider removing the multiple choices.left1050000left450400040465700left241951200Unit 2: Equality, Inequality & Problem SolvingLesson 20: Inequality ProblemsStudent VersionName_______________________Date________________________Opening Activity:You want to join a health club to get back into shape.The Tone Up Club charges $50 to join plus $9 per month.Be in Shape has no sign up fee and charges $12 per month.You have a maximum of $200 to spend. Which club will you join? Why?Activity 2:A) You gave your brother a 50-foot start in a race. If he ran 10 feet every 5 seconds what is the minimum speed you would have to run to catch him in at least 20 seconds from the time you both started?B) Given 2x + ax – 7 > -12, determine the largest integer value of a when x = -1. Unit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson 21: Final Performance Tasks/ProjectTask 1:When working with inequalities we saw that something unusual happened when we multiplied or divided by a negative number. Explain what happens and why it is happening. You can use examples to support your thinking.474345040640Task 2: Birthday RiddleSolving a Situational Unknown in at least two waysAnthony is 4 years older than Kasim. Kasim is twice as old as Ramon. Their ages sum up to 64. How old is each of them?458152570485Task 3: Great AdventuresYour class wants to go on a trip to Great Adventures. There are 25 students in your class. Tickets and bus ride cost $35 a person. You are going to hold a cake sale at school. You were given a donation of cakes, cookie and soda by parents so everything you sell will go towards the trip. You have to decide how much you will sell each item for so that you make the minimum amount of money so that everyone could go on the trip. Come up with possible solutions to this problem.Unit 2: Appendix ATwo Connected Inequality LessonsUnit 2: Equality, Inequality & Problem SolvingLesson Bookmarkstc "Beginning of Lessons" \l 2Lesson A: InequalitiesInequalitiesTeacher GuideGetting to Know InequalitiesOpening Activity:What do the ideas of “equality” and “inequality” mean to you?How do you see these ideas outside of mathematics? How do you see these ideas within mathematics?Note to Teacher: The purpose here is for students to think about these ideas first in a global way then within the context of mathematics. These ideas are inherent to the world students live in so you have a chance to hear how they think about it. This activity is also a set up for the next activity where students are asked to group different statements. Will students think about a group called “equality” and a group called “inequality”?Activity 2:Look at the different statements below. Group the ones you think belong together. Any statement can be in more than one group.The 12 statements below include a couple equality statements and many inequalities. We’ve included the mathematical statement that was intended to be communicated and you should feel free to adjust the language if you think the wording is not clear.Read at the different statements below. Decision Maker: How do you want your students to interact with these 12 statements? Do you want them to be able to move them around, meaning they’ll need to be on slips of paper? Or do you want them to work with them in the list form in which they are provided?You need to be more than 5 ft. tall to go on the ride at Great Adventures. (height > 5 ft)The elevator can hold up to 1500 pounds. (total pounds in elevator ≤ 1500 :: You’ll need to decided what you want to do with the implications of negative weight. Does the context actually turn this into a compound inequality? So 0 < total pounds in elevator ≤ 1500?)Stephanie estimated the cost of the book with tax to be greater than $10.50 but less than $12.00. ($10.50 < price of book < $12.00)John earned $40,000 last year. ( John’s income = $40,000)The sightseeing road trip from NYC to Los Angeles will take at least 7 days but no more than 15 days. ( 7 days ≤ road trip from NYC to Los Angeles < 15 days)The new contract for the basketball player was rumored to earn him more than $5 million, and up to $8 million per season. ( $5 million < $ per season ≤ $8 million)The football was viewed by 60,000 or more people. (football viewers ≥ 60,000)The temperature is not going to be higher than 30° F tomorrow. (temperature tomorrow < 30° F)Lilliana falls in the group of people that earn greater than $50,000 and less than $100,000. ($50,000 < Liliana’s income < $100,000)Miguel is 5’5” tall and Jamal is 5’8” tall. (There are several ways to represent this depending on how the student interrupts it. Among the interruptions the students may have… Miguel’s height = 5’5” and Jamal’s height = 5’8”, or Jamal’s height > than Miguel’s height [how would this be graphed?] )Jorge estimated he was 15 to 20 minutes late. (15 mins ≤ mins Jorge estimated he was late ≤ 20 mins)The GPS device indicated that the upcoming traffic jam was going to more than 30 mins to get through. ( time through traffic jam > 30 mins)Learner: The goal of this activity is to see how students observe relationships. Different answers are possible and they all can be valued. It will be interesting to see if students can separate the statements that represent an inequality from the other statements that represent equalities. If that doesn’t occur be ready to present those two groups and ask the students why you grouped them in that manner. The third activity should occur on a separate page so students would not have it to look at when they do the second activity Also, if you can represent some of the statements in the form of a picture that would be great.Activity 3:Now let us look at these two groups and I want you to think about how you can represent them mathematically.Let’s start with the group we called “equalities”. How would you represent each of them mathematically, without any words?Decision Maker: Give students about one minute and then there can be a quick discussion about what makes them equalities and how we represent them.Now let’s look at the other group. With your group mates decide how to represent each of the events mathematically without any words.Questioner: Discuss the different results. You can ask questions like, “What is the difference between h>5 and h>5?” or “Why wasn’t h>5 appropriate for statement 1?” or “what would you have to change in the first statement to make the result h>5?”We call all these type of statements inequalities. Can you come up with all the possible types of situations that can occur? For example we can have “greater than” represented as c>2. Decision Maker: Have the students share out all the possibilities they can come up with.Activity 4:Let us look at these different symbols. Do any of these have the same meaning? Justify your answer.5464175323850048736253238500370459033020004315460374650032480253810000< < > >=≠< < > >36385505651500303847546990002486025565150018669004699000Note to Teacher: ≠< < > > These are suppose to be NOT equal to, NOT less than, NOT less than or equal to, NOT greater than, and NOT greater than or equal to respectively.Questioner: An interesting discussion can occur about which mean the same. A question that can help kids think about this is: If x < 4 what values of x make that statement true?Activity 5:Now you are going to create two “real world” inequality statements in words that your classmates have to write in algebraic form. See if you can stump your classmates. Decision Maker: Have students present their ideas to each other. An issue can come up if they use ideas that can only be represented as whole numbers, such as number of people. Consider letting them think about if there is an issue with this situation. It will be looked at in an upcoming lesson.Unit 2: Equality, Inequality & Problem SolvingGetting to Know Inequalities Student VersionName_______________________Date________________________Opening Activity:What do the ideas of “equality” and “inequality” mean to you?How do you see these ideas outside of mathematics? How do you see these ideas within mathematics?Activity 2:Look at the different statements below. Group the ones you think belong together. Any statement can be in more than one group.You need to be more than 5 ft. tall to go on the ride at Great Adventures.The elevator can hold up to 1500 pounds.Stephanie estimated the cost of the book with tax to be greater than $10.50 but less than $12.00.John earned $40,000 last year.The sightseeing road trip from NYC to Los Angeles will take at least 7 days but no more than 15 days.The new contract for the basketball player was rumored to earn him more than $5 million, and up to $8 million per season.The football was viewed by 60,000 or more people.The temperature is not going to be higher than 30° F tomorrow.Lilliana falls in the group of people that earn greater than $50,000 and less than $100,000.Miguel is 5’5” tall and Jamal is 5’8” tall.Jorge estimated he was 15 to 20 minutes late.The GPS device indicated that the upcoming traffic jam was going to more than 30 mins to get through. Activity 3:Now let us look at these two groups and I want you to think about how you can represent them mathematically.Let’s start with the group we called “equalities”. How would you represent each of them mathematically, without any words?Now let’s look at the other group. With your group mates decide how to represent each of the events mathematically without any words.We call all these type of statements inequalities. Can you come up with all the possible types of situations that can occur? For example we can have “greater than” represented as c >2.Activity 4:Let us look at these different symbols. Do any of these have the same meaning? Justify your answer.48736253238500370459033020004315460374650032480253810000< < > >=≠< < > >Activity 5:Now you are going to create two “real world” inequality statements in words that your classmates have to write in algebraic form. See if you can stump your classmates. Unit 2: Equality, Inequality & Problem SolvingLesson B: Inequalities ContinuedInequalityTeacher GuideAnother Representation of Inequalities Opening Activity:Look at the following number lines and describe the similarities and differences that you notice. What do you think each of them means? Why?486229279401-10234x001-10234x a.515257882651-10234x001-10234xb.515256339271-10234x001-10234xc. 4717141206501-10234x001-10234xd.Learner: A goal here is to find out if students can transfer their knowledge from the first lesson and transfer it to the new ideas of number lines. Have students share their ideas and help them to discuss the differences and what they think each of them means. Activity 2:How do you think you would represent each of the inequalities we just looked at mathematically? Work with each graph and represent them algebraically.Decision Maker: Have students share and explain their reasoning.Activity 3:Look at the following number lines and describe the similarities and differences that you notice. How do these number lines differ from the previous set of number lines? How will that affect how you represent them algebraically? Then represent them algebraically10668001908631-10234x001-10234xa. 11103431261841-10234x001-10234xb.10668001776191-10234x001-10234xc.10668001855111-10234x001-10234xd. Learner: Again it will be important to see how students transfer their knowledge from the previous lesson. It is important that students can explain their reasoning. This is a new idea in representing them algebraically so students will probably need supportActivity 4:You are going to be given a set of scenarios. Your job is to represent them both algebraically and graphically.You have to give at least five dollars if you want to join the group.In order for the school to hold the prom more a minimum of $500 is needed and the maximum amount needed is $2000.At the trip to the ice skating rink we were told we could stay on the ice for at most three hours.Learner: Again it would be good to have students share their thinking and reasoning. Provocateur: The last one is interesting. Which is a better answer x<3 or 0<x<3?Final Activity: Performance Task- Create a ScenarioCreate a scenario for each of these inequalitiesx < 10011 < x < 190 < x < 1000Another Representation of InequalitiesStudent VersionName_______________________Date________________________ Opening Activity:Look at the following number lines and describe the similarities and differences that you notice. What do you think each of them means? Why? 486229279401-10234x001-10234x a.515257882651-10234x001-10234xb.515256339271-10234x001-10234xc. 4717141206501-10234x001-10234xd. Activity 2:How do you think you would represent each of the inequalities we just looked at mathematically? Work with each graph and represent them algebraically.Activity 3:Look at the following number lines and describe the similarities and differences that you notice. How do these number lines differ from the previous set of number lines? How will that affect how you represent them algebraically? Then represent them algebraicallyActivity 4:You are going to be given a set of scenarios. Your job is to represent them both algebraically and graphically.You have to give at least five dollars if you want to join the group.In order for the school to hold the prom more a minimum of $500 is needed and the maximum amount needed is $2000.At the trip to the ice skating rink we were told we could stay on the ice for at most three hours.Unit 2: Equality, Inequality & Problem SolvingFinal Activity: Performance Task- Create a ScenarioStudent VersionName_______________________Date________________________Create a scenario for each of these inequalitiesx < 10011 < x < 190 < x < 1000 ................
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