Overview. Essential Questions. Algebra I, Quarter 1, Unit 1.2 Interpreting and Applying Algebraic Expressions


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1 Algebra I, Quarter 1, Unit 1.2 Interpreting and Applying Algebraic Expressions Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Write and interpret an expression from a given context. Interpret the parts of an expression. Interpret complicated expressions by decomposing the expressions into separate parts. Identify ways to rewrite an expression. Understand that polynomials are closed under the operations of addition, subtraction, and multiplication. Mathematical Practices to Be Integrated 2 Reason abstractly and quantitatively. Use algebraic properties and chunking to rewrite expressions. Use arithmetic operations to simplify expressions. 4 Model with mathematics. Interpret expressions from a context 5 Use appropriate tools strategically. Use the evaluation mode on a graphing calculator to evaluate expressions. Use the test function on a graphing calculator to show the relationship between two expressions. Essential Questions How do you show that two algebraic expressions are equivalent? Why are degree, coefficients, terms, factors, and variables, important in the study of polynomials? Which arithmetic properties can be used to evaluate expressions? Under which arithmetic operations are polynomials a closed set? Providence Public Schools D8
2 Interpreting and Applying Algebraic Expressions (6 8 days) Algebra 1, Quarter 1, Unit 1.2 Standards Common Core State Standards for Mathematical Content Algebra Seeing Structure in Expressions ASSE Interpret the structure of expressions [Linear, exponential, quadratic] ASSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. ASSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Arithmetic with Polynomials and Rational Expressions AAPR Perform arithmetic operations on polynomials [Linear and quadratic] AAPR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. Common Core State Standards for Mathematical Practice 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. 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 are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, 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 Providence Public Schools D9
3 Algebra 1, Quarter 1, Unit 1.2 Interpreting and Applying Algebraic Expressions (6 8 days) reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know 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, twoway tables, graphs, flowcharts and formulas. They can 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. 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. Clarifying the Standards Prior Learning In Grade 6, students identified parts of an expression containing whole numbers. They applied the properties of operations to generate equivalent expressions. Students also identified when two expressions are equivalent (for example, y + y+ y = 3y). In Grade 7, students applied properties of operations as strategies to add, subtract, factor, and expand linear expressions. They also understood that rewriting an expression in different forms can show how quantities are related. In Grade 8, students applied the properties of integer exponents to generate equivalent numerical expressions. D10 Providence Public Schools
4 Interpreting and Applying Algebraic Expressions (6 8 days) Algebra 1, Quarter 1, Unit 1.2 Current Learning Performing arithmetic operations on polynomials is a fluency standard for Algebra I. Students become fluent in adding, subtracting, and multiplying polynomials, which will support their work throughout the study of Algebra. Being able to identify structure in expressions by viewing one or more parts as single entities is also a fluency standard in Algebra I. Students become fluent in transforming expressions and chunking, which are essential to factoring later in Algebra I. Students interpret the parts of simple and more complicated expressions from a given context, which must be modeled by students. They simplify and rewrite expressions by combing like terms, using properties as a means to accomplish this objective. Students also learn and understand that polynomials are closed within the operations of addition, subtraction, and multiplication. Future Learning In Algebra II, students will be interpreting the structure of polynomial and rational expressions. Also, they will take their ability to perform arithmetic operations on polynomials and extend it beyond quadratics. In addition, students will learn how to divide polynomials to determine the zeros of the function. Students will use polynomial identities to describe numerical relationships. Additional Findings Students have difficulties thinking of algebraic expressions and variables to represent a range of values. Adding It Up describes a study designed to address these difficulties, in which students were asked to give instructions to an idealized mathematics machine. The students easily made sense of the idea of employing letters to write rules that would enable the machine to solve whole classes of problems. (p. 264) Assessment When constructing an endofunit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standardsbased mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performancebased tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Write verbal expressions and algebraic expressions Use the order of operations to evaluate numeric and algebraic expressions Identify the commutative, associative, equality and identity properties Use the Distributive Property to simplify expressions Write polynomials in standard form Show that two algebraic expressions are equivalent Use the graphing calculator to simplify expressions Providence Public Schools D11
5 Algebra 1, Quarter 1, Unit 1.2 Interpreting and Applying Algebraic Expressions (6 8 days) Instruction Learning Objectives Students will be able to: Identify and interpret parts of expressions Use symbols to represent unknowns and variables Write verbal expressions for algebraic expressions Write algebraic expressions for verbal expressions Evaluate numeric and algebraic expressions using the order of operations Identify properties of numbers (commutative, associative, equality and identity) Use the Distributive Property to simplify expressions Find the degree of a polynomial and write them in standard form Use evaluation mode on the graphing calculator to evaluate expressions Use test function on the graphing calculator to show the relationship between two expressions Demonstrate understanding of concepts and skills learned in this unit. Resources From Algebra 1, (Glencoe McGraw Hill) 2010 Section 1.1 pp. 5 to 9 Section 1.2 pp. 6 to 15 Section 1.3 pp. 16 to 22 Section 1.4 pp. 23 to 29 Section 7.4 pp. 424 to 429 Chapter 1 Resource Masters of corresponding sections Chapter 7 Resource Masters of corresponding sections 5Minute Check Transparencies for corresponding sections Quick Review Math Handbook, (Glencoe McGraw Hill) 2010 Sections 6.1, 6.2, 7.5 (for use with Distributive Property) Interactive Classroom CD: PowerPoint presentations Teacher Works Plus CDROM Teaching with Foldables (Dinah Zike; Glencoe McGraw Hill 2010) Exam View Assessment Suite Software Math Online, glencoe.com Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. D12 Providence Public Schools
6 Interpreting and Applying Algebraic Expressions (6 8 days) Algebra 1, Quarter 1, Unit 1.2 Materials Algebra tiles, TINspire calculators Instructional Considerations Key Vocabulary degree of polynomial Planning for Effective Instructional Design and Delivery The reinforced vocabulary taught in previous grades or units includes numerical expression, algebraic expression, coefficient, like terms, variable, exponent, base, power, reciprocal. The focus of this unit is algebraic expressions. Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) (ASSE.1b) is the foundation for students future work in factoring, completing the square, and other algebraic calculations. Fluency in adding, subtracting, and multiplying polynomials (AAPR.1) supports students throughout algebra, as well as their work with functions. It is important for students to understand that mathematics is a language and that symbols represent words. Students should have opportunities to explore the different meanings used in algebraic expressions that represent, addition, subtraction, multiplication and division. Students should become fluent in translating between numerical and written, and algebraic and written expressions. The section on math practices refers to a practice called chunking. Chunking is a way to highlight the similarity of expressions. It allows the students to see a whole expression as many parts. Chunking can be useful in teaching the distributive property. For example, 7 * 49 can be rewritten as 7 * (501). The students now see this as which is much easier to simplify. Chunking also works with algebraic expressions. For example, 17m + 25m can be rewritten as (17+25) m. Rewriting it in this way may help students better understand combining like terms. Chunking will help some students better understand future work with linear quadratic, and exponential expressions. Incorporate the Essential Questions as part of the daily lesson. Options include using them as a do now to activate prior knowledge of the previous day s lesson, using them as an exit ticket by having students respond to it and post it, or hand it in as they exit the classroom, or using them as other formative assessments. Essential questions should be included in the unit assessment. The 5minute check transparencies can be used as a cues, questions, and advance organizers strategy as students will be activating prior knowledge. Some 5minute checks may take longer than the allotted time, so consider choosing only problems that activate prior knowledge and use the rest for differentiation, to formatively assess student learning, as an exit ticket, or assigning for homework. For planning considerations read through the teacher edition for suggestions about scaffolding techniques, using additional examples, and differentiated instructional guidelines as suggested by the Glencoe resource. Providence Public Schools D13
7 Algebra 1, Quarter 1, Unit 1.2 Interpreting and Applying Algebraic Expressions (6 8 days) Nonlinguistic representations including graphic organizers and physical models such as algebra tiles would be appropriate for teaching distributive property. Algebra tiles provide conceptual understanding of combining like terms. Algebra tiles and area models also help students understand the distributive property through manipulating the tiles and help to identify strategies to write expressions in different ways. For example: 3 x 7 = 3(x) 3(7) = 3x 21 Students should also be given opportunities to write and interpret expressions from given context. The Chapter One Resource Masters book and various problems within sections 1.1 to 1.4 provide several opportunities for students to practice writing and interpreting expressions. When writing and evaluating expressions, students should be able to identify equivalent expressions without necessarily evaluating them. You can create a matching game by writing equivalent expressions on different index cards and having students match equivalent expressions. This activity will help you quickly access students fluency with manipulating terms in an expression. Students should be able to name, distinguish, and label the parts that make up an expression, such as coefficients, terms, factors, variables, and exponents. When working with order of operations, you may use the TINspire Calculator so that students can become familiar with the key locations. You may use the transparencies provided in the ancillary material as focus activities, review, or an exit activity. Additional activities and examples may be used for homework assignments. Notes D14 Providence Public Schools
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