Gary School Community Corporation Mathematics Department Unit Document Unit Number: 1 Grade: Algebra 1 Unit Name: Polynomial Operations (Add & Sub) Duration of Unit: A1.RNE.7 Standards for Mathematical Content UNIT FOCUS ***** Standard Emphasis Critical Important Additional Understand polynomials are closed under the operations of addition, subtraction, and multiplication with integers; add, subtract, and multiply polynomials and divide polynomials by monomials. A1.RNE.1 Understand the hierarchy and relationships of numbers and sets of numbers within the real number system. ***** A1.RNE.2 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational ***** Mathematical Process Standards: PS.1: Make sense of problems and persevere in solving them. PS.2: Reason abstractly and quantitatively PS.3: Construct viable arguments and critique the reasoning of others PS.4: Model with mathematics PS.5: Use appropriate tools strategically PS.6: Attend to Precision PS.7: Look for and make use of structure PS.8: Look for and express regularity in repeated reasoning ******** Big Ideas/Goals Essential Questions/ Learning Targets I Can Statements Vertical Articulation documents for K 2, 3 5, and 6 8 can be found at: http://www.doe.in.gov/standards/mathematics (scroll to bottom)
Multiple polynomials can be simplified when adding or subtracting. A set is closed under operations if the operation on two elements of the set produces another element of the set. How do we add and subtract polynomials? How do we group like terms? How do we apply the properties to simplify expressions? What do we mean when we say a set is closed under an operation? I can add and subtract polynomials. I can group like terms. I can apply properties to simplify expressions. I can explain what it means of a set to be closed or not closed under an operation. I can give examples of what it means for a set to be closed or not closed under an operation. Within the Real Number System, numbers are classified as rational or irrational. Rational numbers consist of several subsets. Results of sums, and products within the real number system will follow a pattern How do we classify rational or irrational numbers? How would we describe the relationships among natural numbers, whole numbers, and integers? What are the subsets of rational numbers? How would we determine whether the sum or product of rational and irrational numbers? I can categorize the Real Number System. I can describe the relationships among natural numbers, whole numbers, and integers. I can explain that the sum of two rational numbers is rational. I can explain that the product of two rational numbers is rational. I can explain that the sum of a rational number and an irrational number is irrational. I can explain that the product of a nonzero rational number and an irrational number is irrational. UNIT ASSESSMENT TIME LINE 2
Beginning of Unit Pre-Assessment Assessment Name: Revisiting Rational and Irrational Numbers Assessment Type: Short Answer Assessment Standards: 8.NS.1, 8NS.2 Assessment Description: See Attachment Throughout the Unit Formative Assessment Assessment Name: Understanding the concept of Closure Assessment Type: Think-Pair-Write Students are given a math topic/question, allow time for students to individually process their thinking to write a response, then have them share with their response with their partners. Are polynomials closed under addition? Prove you answer. Assessment Standards: A1.RNE.7 Assessment Description: Students collaborate to consider the concept of closure (A set is closed under operations if the operation on two elements of the set produces another element of the set.) Example: If you add two even numbers (from the set of even numbers), is the sum even? Checking: 8 + 12 = 20 Yes, 20 is even. 18 + 8 = 26 Yes, 26 is even. 8 + 100 = 102 Yes, 108 is even. Since the sum (the answer) is always even, the set of even numbers is closed under the operation of addition. Let's check out this question. If you divide two even numbers (from the set of even numbers), is the quotient (the answer) even? Checking: 24 / 6 = 4 Yes, 4 is even. 24 / 3 = 8 Yes, 8 is even. 100 / 4 = 25 NO, 25 is not even! When you find even ONE example that does not work, the set is not closed under that operation. The even numbers are not closed under division. 3
Assessment Name: Sums and Products of Real Numbers Assessment Type: Sentence Starters- Start the sentence and have students finish it. One thing I learned today The sum or product of two rational numbers is The sum or product of a rational number and an irrational number is Assessing Standards: A1.RNE.2 Assessment Description: Students recognize and demonstrate that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Assessment Name: Closure and Polynomials Assessment Type: Open-Ended - Open-ended problem is a problem that has several or many correct answers, and several ways to the correct answer(s). Assessing Standards: A1.RNE.7 Assessment Description: Understand polynomials are closed under the operations of addition, subtraction, and multiplication with integers; add, subtract, and multiply polynomials and divide polynomials by monomials. Example: Write 5 polynomials with various degrees. (Let the class make-up 5) Determine whether polynomials are closed under the following properties: Addition. Choose any two of the above polynomials and add them, it remains a polynomial so polynomials are closed under addition. Also, graph the sum of the two polynomials on your calculator. Does the graph remain smooth and continuous? If yes, then it s a polynomial. Subtraction. Choose any two of the above polynomials and subtract them, it remains a polynomial so polynomials are closed under subtraction. Also, graph the difference of the two polynomials on your calculator. Does the graph remain smooth and continuous? If yes, then it s a polynomial. Multiplication. Choose any two of the above polynomials and multiply them; it remains a polynomial so polynomials are closed under multiplication. Also, graph the product of the two polynomials on your calculator. Does the graph remain smooth and continuous? If yes, then it s a polynomial. Division. Choose any two of the above polynomials and divide them, it DOES NOT remain a polynomial so polynomials are NOT closed under division. Also, graph the quotient of the two polynomials on your calculator. Does the graph remain smooth and continuous? If not, then it s not a polynomial. 4
End of Unit Summative Assessments Assessment Name: TBD Assessment Type: End of Unit Assessing Standards: A1.RNE.7, A1.RNE.1, A1.RNE.2 Assessment Description: PLAN FOR INSTRUCTION Unit Vocabulary Key terms are those that are newly introduced and explicitly taught with expectation of student mastery by end of unit. Prerequisite terms are those with which students have previous experience and are foundational terms to use for differentiation. Key Terms for Unit Expressions Distributive Property Like Terms Operations of Integers Coefficient Exponents (Laws of Exponents) Monomial Polynomial Real Number System Rational Numbers Irrational Numbers Variable Set Element Prerequisite Math Terms Unit Resources/Notes Include district and supplemental resources for use in weekly planning For more information on Closure and Polynomials Assessment, go to: http://www.davis.k12.ut.us/cms/lib09/ut01001306/centricity/domain/7586/secondary%20iii/sec%20iii% 20term%201/Unit%203%20Study%20Notes%204.pdf http://www.softschools.com GOOD WEBSITES FOR MATHEMATICS: http://nlvm.usu.edu/en/nav/vlibrary.html http://www.math.hope.edu/swanson/methods/applets.html 5
http://learnzillion.com http://illuminations.nctm.org https://teacher.desmos.com http://illustrativemathematics.org http://www.insidemathematics.org https://www.khanacademy.org/ https://www.teachingchannel.org/ http://map.mathshell.org/materials/index.php https://www.istemnetwork.org/index.cfm http://www.azed.gov/azccrs/mathstandards/ Targeted Process Standards for this Unit PS.1: Make sense of problems and persevere in solving them Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway, rather than simply jumping into a solution attempt. They consider analogous problems and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? and "Is my answer reasonable?" They understand the approaches of others to solving complex problems and identify correspondences between different approaches. Mathematically proficient students understand how mathematical ideas interconnect and build on one another to produce a coherent whole. PS.2: Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. PS.3: Construct viable arguments and critique the reasoning of others Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They analyze situations by breaking them into cases and recognize and use counterexamples. They organize their mathematical thinking, justify their conclusions and communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. They justify whether a given statement is true always, sometimes, or never. Mathematically proficient students participate and collaborate in a mathematics community. They listen to or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 6
PS.4: Model with mathematics Mathematically proficient students apply the mathematics they know to solve problems arising in everyday life, society, and the workplace using a variety of appropriate strategies. They create and use a variety of representations to solve problems and to organize and communicate mathematical ideas. Mathematically proficient students apply what they know and are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. PS.5: Use appropriate Tools Strategically Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. PS.6: Attend to precision Mathematically proficient students communicate precisely to others. They use clear definitions, including correct mathematical language, in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They express solutions clearly and logically by using the appropriate mathematical terms and notation. They specify units of measure and label axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently and check the validity of their results in the context of the problem. They express numerical answers with a degree of precision appropriate for the problem context. PS.7: Look for and make use of structure Mathematically proficient students look closely to discern a pattern or structure. They step back for an overview and shift perspective. They recognize and use properties of operations and equality. They organize and classify geometric shapes based on their attributes. They see expressions, equations, and geometric figures as single objects or as being composed of several objects. PS.8: Look for and express regularity in repeated reasoning Mathematically proficient students notice if calculations are repeated and look for general methods and shortcuts. They notice regularity in mathematical problems and their work to create a rule or formula. Mathematically proficient students maintain oversight of the process, while attending to the details as they solve a problem. They continually evaluate the reasonableness of their intermediate results. 7