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Algebra 1 Unit B: Equations and Inequalities Math Florida Standards Unit Overview Content Standards (bold are directly assessed) You can represent an equation/inequality in many ways. You can use properties of numbers and equality to transform equations/inequalities into equivalent, simpler equations/inequalities and find solutions. Real-world relationships can be modeled using equations/inequalities. In a proportional relationship the relationship between two quantities are equal. You can use this relationship to describe similar figures, scale models, and rates. Textbook Resources Pearson Prentice Hall Algebra 1 copyright 2011 Pearson SuccessNet Sections: 1.8, 2.1 – 2.5, 2.5 Concept Byte, 2.6 – 2.8, 2.10, 3.2 – 3.4, 3.6, 3.7 Mathematics Formative Assessment System The system includes tasks or problems that teachers can implement with their students, and rubrics that help the teacher interpret students' responses. Teachers using MFAS ask students to perform mathematical tasks, explain their reasoning, and justify their solutions. Rubrics for interpreting and evaluating student responses are included so that teachers can differentiate instruction based on students' strategies instead of relying solely on correct or incorrect answers. The objective is to understand student thinking so that teaching can be adapted to improve student achievement of mathematical goals related to the standards. Like all formative assessment, MFAS is a process rather than a test. Research suggests that well-designed and implemented formative assessment is an effective strategy for enhancing student learning. MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.4 MAFS.912.N-Q.1.1* MAFS.912.N-Q.1.2* MAFS.912.N-Q.1.3* *(all N-Q are assessed throughout) MAFS.912.A-SSE.1.1b MAFS.912.A-REI.1.1 MAFS.912.A-REI.2.3 Highlighted Standards for Mathematical Practice MAFS.K12.MP.1.1 MAFS.K12.MP.2.1 MAFS.K12.MP.4.1 MAFS.K12.MP.6.1 MAFS.K12.MP.8.1 Other Resources Algebra Nation Online Graphing Calculator Canvas Mathematics Website Algebra Balance Scales Paying the Rent Planes and Wheat Mathematics Formative Assessment System Tasks This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 1 Unit B: Equations and Inequalities Unit Scale (Multidimensional) (MDS) The multidimensional, unit scale is a curricular organizer for PLCs to use to begin unpacking the unit. The MDS should not be used directly with students and is not for measurement purposes. This is not a scoring rubric. Since the MDS provides a preliminary unpacking of each focus standard, it should prompt PLCs to further explore question #1, “What do we expect all students to learn?” Notice that all standards are placed at a 3.0 on the scale, regardless of their complexity. A 4.0 extends beyond 3.0 content and helps students to acquire deeper understanding/thinking at a higher taxonomy level than represented in the standard (3.0). It is important to note that a level 4.0 is not a goal for the academically advanced, but rather a goal for ALL students to work toward. A 2.0 on the scale represents a “lightly” unpacked explanation of what is needed, procedural and declarative knowledge i.e. key vocabulary, to move students towards proficiency of the standards. 4.0 In addition to displaying a 3.0 performance, the student must demonstrate in-depth inferences and applications that go beyond what was taught within these standards. Examples: Create a word problem involving a real-life relationship. Choose and define appropriate variables and use them to create an equation that models the relationship. Solve the equation, justifying each step, and explain its meaning in the context of the problem. 3.0 The Student will: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. (MAFS.912.A-CED.1.1) Rearrange 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. (MAFS.912.A-CED.1.4) Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. (MAFS.912.N-Q.1.1) Define appropriate quantities for the purpose of descriptive modeling. (MAFS.912.N-Q.1.2) Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. (MAFS.912.N-Q.1.3) Interpret expressions that represent a quantity in terms of its context. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret as the product of P and a factor not depending on P. (MAFS.912.A-SSE.1.1b) 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. Construct a viable argument to justify a solution method. (MAFS.912.A-REI.1.1) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. (MAFS.912.A-REI.2.3) 2.0 The student will recognize or recall specific vocabulary, such as: Linear equation, linear inequality, properties of equality for real numbers, at most, at least, more than, less than, number line The student will perform basic processes, such as: Solving a linear equation and an inequality, graphing on a number line 1.0 With help, partial success at 2.0 content but not at score 3.0 content This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 1 Unit B: Equations and Inequalities Unpacking the Standard: What do we want students to Know, Understand and Do (KUD): The purpose of creating a Know, Understand, and Do Map (KUD) is to further the unwrapping of a standard beyond what the MDS provides and assist PLCs in answering question #1, “What do we expect all students to learn?” It is important for PLCs to study the focus standards in the unit to ensure that all members have a mutual understanding of what student learning will look and sound like when the standards are achieved. Additionally, collectively unwrapping the standard will help with the creation of the uni-dimensional scale (for use with students). When creating a KUD, it is important to consider the standard under study within a K-12 progression and identify the prerequisite skills that are essential for mastery. Domain: Algebra: Reasoning with Equations & Inequalities Cluster: Understand solving equations as a process of reasoning and explain the reasoning- (Major Cluster) Standard: 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. Construct a viable argument to justify a solution method. (MAFS.912.A.REI.1.1) Understand “Essential understandings,” or generalizations, represent ideas that are transferable to other contexts. When or why steps are completed in a certain order. Arguments about equality are used to justify solutions to equations. There is a justifiable process for solving equations in which the equality of expressions must be maintained. Know Declarative knowledge: Facts, vocab., information steps for solving solution no solution addition property of equality subtraction property of equality multiplication property of equality division property of equality distributive property like terms Do Procedural knowledge: Skills, strategies and processes that are transferrable to other contexts. Retrieval Identify properties of equality Comprehension Explain each step Analysis Construct a viable argument to justify solution Prerequisite skills: What prior knowledge (foundational skills) do students need to have mastered to be successful with this standard? inverse operations, expressions, order of operations, variables Moving Beyond: Solving polynomial and trigonometric equations This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 1 Unit B: Equations and Inequalities Uni-Dimensional, Lesson Scale: The uni-dimensional, lesson scale unwraps the cognitive complexity of a focus standard for the unit, using student friendly language. The purpose is to articulate distinct levels of knowledge and skills relative to a specific topic and provide a roadmap for designing instruction that reflects a progression of learning. The sample performance scale shown below is just one example for PLCs to use as a springboard when creating their own scales for student-owned progress monitoring. The lesson scale should prompt teams to further explore question #2, “How will we know if and when they’ve learned it?” for each of the focus standards in the unit and make connections to Design Question 1, “Communicating Learning Goals and Feedback” (Domain 1: Classroom Strategies and Behaviors). Keep in mind that a 3.0 on the scale indicates proficiency and includes the actual standard. A level 4.0 extends the learning to a higher cognitive level. Like the multidimensional scale, the goal is for all students to strive for that higher cognitive level, not just the academically advanced. A level 2.0 outlines the basic declarative and procedural knowledge that is necessary to build towards the standard. Standard: 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. Construct a viable argument to justify a solution method. (MAFS.912.A.REI.1.1) Learning Progression Sample Tasks Score 4.0 3.5 3.0 2.5 I can… Given a simple equation such as -3(2x - 5) = 39 , generate multiple solution methods and construct viable arguments to justify each solution method. Experiment with a simple equation to generate multiple solution methods and construct viable arguments to justify each solution method Investigate errors in solution methods given an equation and multiple solutions I can do everything at a 3.0, and I can demonstrate partial success at score 4.0. I can… Solve the following equation and justify each step: Explain each step in solving a simple equation Construct a viable argument to justify the solution I can do everything at a 2.0, and I can demonstrate partial success at score 3.0. I can… Solve a simple equation Identify properties of equality Recognizing a viable argument Solve the following equation: 2 4 x - 2 = - (x + 3) 3 3 5x + 2 = 2(x + 7) Which property of equality is demonstrated as the first step in solving the following equation? 2.0 Key Vocabulary such as: solution, no solution, simple equation, addition property of equality, subtraction property of equality, multiplication property of equality, division property of equality, distributive property, like terms 1.0 7 k +1 = 2k - 5 2 7 2( k +1) = 2(2k - 5) 2 I need prompting and/or support to complete 2.0 tasks. This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015 Algebra 1 Unit B: Equations and Inequalities Sample High Cognitive Demand Tasks: These task/guiding questions are intended to serve as a starting point, not an exhaustive list, for the PLC and are not intended to be prescriptive. Tasks/guiding questions simply demonstrate one way to help students learn the skills described in the standards. Teachers can select from among them, modify them to meet their students’ needs, or use them as an inspiration for making their own. They are designed to generate evidence of student understanding and give teachers ideas for developing their own activities/tasks and common formative assessments. These guiding questions should prompt the PLC to begin to explore question #3, “How will we design learning experiences for our students?” and make connections to Marzano’s Design Question 2, “Helping Students Interact with New Knowledge”, Design Question 3, “Helping Students Practice and Deepen New Knowledge”, and Design Question 4, “Helping Students Generate and Test Hypotheses” (Domain 1: Classroom Strategies and Behaviors). Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic MAFS Content Standard(s) functions, and simple rational, absolute, and exponential functions. (MAFS.912.A-CED.1.1) Design Question 1; Element 1 MAFS Mathematical Practice(s) Make sense of problems and persevere in solving them. (MAFS.K12.MP.1.1) Attend to precision. (MAFS.K12.MP.6.1) Design Question 1; Element 1 Marzano’s Taxonomy Knowledge Utilization Teacher Notes Questions to develop mathematical thinking, possible misconceptions/misunderstandings , how to differentiate/scaffold instruction, anticipate student problem solving strategies Task *These tasks can either be teacher created or modified from a resource to promote higher order thinking skills. Please cite the source for any tasks. Licensed by Illustrative Mathematics under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License This a working document that will continue to be revised and improved taking your feedback into consideration. Pasco County Schools, 2014-2015