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STANDARDS FOR MATHEMATICS High School Algebra 1 1 High School Overview Conceptual Categories and Domains Number and Quantity The Real Number System (N-RN) Quantities (N-Q) The Complex Number System (N-CN) Vector and Matrix Quantities (N-VM) Algebra Seeing Structure in Expressions (A-SSE) Arithmetic with Polynomials and Rational Expressions (A-APR) Creating Equations (A-CED) Reasoning with Equations and Inequalities (A-REI) Functions Contemporary Mathematics Discrete Mathematics (CM-DM) Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models (F-LE) Trigonometric Functions (F-TF) Geometry Statistics and Probability Interpreting Categorical and Quantitative Data (SID) Making Inferences and Justifying Conclusions (SIC) Conditional Probability and the Rules of Probability (S-CP) Using Probability to Make Decisions (S-MD) Congruence (G-CO) Similarity, Right Triangles, and Trigonometry (G-SRT) Circles (G-C) Expressing Geometric Properties with Equations (G-GPE) Geometric Measurement and Dimension (G-GMD) Modeling with Geometry (G-MG) Modeling 2 Domain and Clusters High School - Number and Quantity Overview The Real Number System (N-RN) Extend the properties of exponents to rational exponents Use properties of rational and irrational numbers. Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Quantities (N-Q) Reason quantitatively and use units to solve problems The Complex Number System (N-CN) Perform arithmetic operations with complex numbers Represent complex numbers and their operations on the complex plane Use complex numbers in polynomial identities and equations Vector and Matrix Quantities (N-VM) Represent and model with vector quantities. Perform operations on vectors. Perform operations on matrices and use matrices in applications. 3 High School - Algebra Overview Seeing Structure in Expressions (A-SSE) Interpret the structure of expressions Write expressions in equivalent forms to solve problems Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Arithmetic with Polynomials and Rational Expressions (A-APR) Perform arithmetic operations on polynomials Understand the relationship between zeros and factors of polynomials Use polynomial identities to solve problems Rewrite rational expressions Creating Equations (A-CED) Create equations that describe numbers or relationships Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Solve systems of equations Represent and solve equations and inequalities graphically 4 High School - Functions Overview Interpreting Functions (F-IF) Understand the concept of a function and use function notation Interpret functions that arise in applications in terms of the context Analyze functions using different representations Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Building Functions (F-BF) Build a function that models a relationship between two quantities Build new functions from existing functions Linear, Quadratic, and Exponential Models (F-LE) Construct and compare linear, quadratic, and exponential models and solve problems Interpret expressions for functions in terms of the situation they model Trigonometric Functions (F-TF) Extend the domain of trigonometric functions using the unit circle Model periodic phenomena with trigonometric functions Prove and apply trigonometric identities 5 High School – Geometry Overview Congruence (G-CO) Experiment with transformations in the plane Understand congruence in terms of rigid motions Prove geometric theorems Make geometric constructions Geometric Measurement and Dimension (G-GMD) Explain volume formulas and use them to solve problems Visualize relationships between two-dimensional and threedimensional objects Modeling with Geometry (G-MG) Apply geometric concepts in modeling situations Similarity, Right Triangles, and Trigonometry (G-SRT) Understand similarity in terms of similarity transformations Prove theorems involving similarity Define trigonometric ratios and solve problems involving right triangles Apply trigonometry to general triangles Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Circles (G-C) Understand and apply theorems about circles Find arc lengths and areas of sectors of circles Expressing Geometric Properties with Equations (G-GPE) Translate between the geometric description and the equation for a conic section Use coordinates to prove simple geometric theorems algebraically 6 High School – Statistics and Probability Overview Interpreting Categorical and Quantitative Data (S-ID) Summarize, represent, and interpret data on a single count or measurement variable Summarize, represent, and interpret data on two categorical and quantitative variables Interpret linear models Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Making Inferences and Justifying Conclusions (S-IC) Understand and evaluate random processes underlying statistical experiments Make inferences and justify conclusions from sample surveys, experiments and observational studies Conditional Probability and the Rules of Probability (S-CP) Understand independence and conditional probability and use them to interpret data Use the rules of probability to compute probabilities of compound events in a uniform probability model Using Probability to Make Decisions (S-MD) Calculate expected values and use them to solve problems Use probability to evaluate outcomes of decisions High School – Contemporary Mathematics Overview Discrete Mathematics (CM-DM) Understand and apply vertex-edge graph topics 7 High School - Modeling Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data. A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional cylinder, or whether a two-dimensional disk works well enough for our purposes. Other situations—modeling a delivery route, a production schedule, or a comparison of loan amortizations—need more elaborate models that use other tools from the mathematical sciences. Realworld situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them is appropriately a creative process. Like every such process, this depends on acquired expertise as well as creativity. Some examples of such situations might include: • Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people, and how it might be distributed. • Planning a table tennis tournament for 7 players at a club with 4 tables, where each player plays against each other player. • Designing the layout of the stalls in a school fair so as to raise as much money as possible. • Analyzing stopping distance for a car. • Modeling savings account balance, bacterial colony growth, or investment growth. • Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport. • Analyzing risk in situations such as extreme sports, pandemics, and terrorism. • Relating population statistics to individual predictions. In situations like these, the models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have? The range of models that we can create and analyze is also constrained by the limitations of our mathematical, statistical, and technical skills, and our ability to recognize significant variables and relationships among them. Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for understanding and solving problems drawn from different types of real-world situations. One of the insights provided by mathematical modeling is that essentially the same mathematical or statistical structure can sometimes model seemingly different situations. Models can also shed light on the mathematical structures themselves, for example, as when a model of bacterial growth makes more vivid the explosive growth of the exponential function. 8 The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on these relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle. In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model—for example, graphs of global temperature and atmospheric CO2 over time. Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are empirically based; for example, exponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from a constant reproduction rate. Functions are an important tool for analyzing such problems. Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are powerful tools that can be used to model purely mathematical phenomena (e.g., the behavior of polynomials) as well as physical phenomena. Modeling Standards Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). 9 Standards for Mathematical Practice: High School Standards for Mathematical Practice Standards Students are expected to: HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. Explanations and Examples High school students start to examine problems 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. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects. 10 Standards for Mathematical Practice Standards Students are expected to: HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. Explanations and Examples High school 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. High school 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. High school students learn to determine domains, to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. 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. High school students 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. 11 Standards for Mathematical Practice Standards Students are expected to: HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. HS.MP.7. Look for and make use of structure. Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. Explanations and Examples High school 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. High school students should be 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, 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. They 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. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 2 By high school, students look closely to discern a pattern or structure. In the expression x + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being 2 composed of several objects. For example, they can see 5 – 3(x – y) as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures. 12 Standards for Mathematical Practice Standards Students are expected to: HS.MP.8. Look for and express regularity in repeated reasoning. Mathematical Practices are listed throughout the grade level document in the 2nd column to reflect the need to connect the mathematical practices to mathematical content in instruction. Explanations and Examples High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 2 3 2 1)(x + x + 1), and (x – 1)(x + x + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 13 High School Algebra 1 Conceptual Category: Number and Quantity (2 Domains, 3 Clusters) Domain: Real Number System (2 Clusters) The Real Number System (N-RN) (Domain 1 - Cluster 2 - Standard 3) Use properties of rational and irrational numbers. Essential Concepts When you perform an operation with two rational numbers you will produce a rational number. When you perform an operation with a nonzero rational and an irrational number you will produce an irrational number. HS.N-RN.B.3 HS.N-RN.B.3. Explain why the sum or product of two rational numbers are 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. Connection: 9-10.WHST.1e Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. Essential Questions Explain what type of number is produced and why when each of the four arithmetic operations is performed on two rational numbers. Explain what type of number is produced and why when each of the four operations is performed on a rational number and an irrational number. Examples & Explanations Since every difference is a sum and every quotient is a product, this includes differences and quotients as well. Explaining why the four operations on rational numbers produce rational numbers can be a review of students understanding of fractions and negative numbers. Explaining why the sum of a rational and an irrational number is irrational, or why the product is irrational, includes reasoning about the inverse relationship between addition and subtraction (or between multiplication and addition). Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2. Example: Explain why the number 2π must be irrational, given that π is irrational. Answer: if 2π were rational, then half of 2π would also be rational, so π would have to be rational as well. Additional Domain Information – The Real Number System (N-RN) Key Vocabulary Rational number Irrational number 14 Rational exponents Example Resources Books Building Powerful Numeracy for Middle and High School Students, by Pamela Weber Harris. http://www.classzone.com/cz/find_book.htm?tmpState=AZ&disciplineSchool=ma_hs&state=AZ&x=24&y=24 This contains supplementary resources for the Arizona adopted math books. Technology http://nlvm.usu.edu/en/nav/topic_t_2.html - Provides teachers or students with virtual manipulatives to interact with the concepts. http://www.khanacademy.org/ - Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. http://www.classzone.com/books/algebra_1/page_build.cfm?content=lesson8_kh_ch11&ch=11 Help with graphing calculators and rational functions. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons For Fractional Exponents: http://www.khanacademy.org/math/algebra/exponents-radicals/v/radical-equivalent-to-rational-exponents This is a short video lesson on converting radicals to fractional exponent notation. http://www.purplemath.com/modules/exponent5.htm This lesson is a good basic introduction to the concept with limited examples. It contains a useful technology extension on using calculators for building conceptual understanding of rational exponents. http://www.themathpage.com/alg/rational-exponents.htm#fractional This lesson includes laws of exponents but moves on to basic equations -3 with rational exponents of the form x5 = 1 8 For Operations Bringing Together Rational and Irrational Numbers: http://www.schooltube.com/video/b5ad397dc525a3795373/ This provides the steppingstones for understanding how adding or subtracting rational and irrational numbers yields an irrational answer. Common Student Misconceptions 1 Students may see a fractional exponent and multiply it by the base. For example, students might say 1 1 27 3 = 27· = 9 instead of 27 3 = 3 27 = 3 . 3 Students may see a negative exponent and do the same, converting the base to a negative number instead of a fraction. 15 Students may have difficulty converting between radical notation and fractional exponent notation: b -m n = and the n. Students tend to assume that they can combine integers and radical expressions: Example, 1 n bm . They might confuse the m 3+ 3 becoming 6 . Students conversely don’t apply available laws of exponents when multiplying or dividing radicals: Example understand that they can split up one radical into the product of two component radicals. 18 3 = 6 . They don’t Domain: Quantities (1 Cluster) Quantities (N-Q) (Domain 2 - Cluster 1 - Standards 1, 2 and 3) Reason quantitatively and use units to solve problems. (Foundation for work with expressions, equations and functions.) Essential Concepts Units and unit relationships can be used to set up and solve multi-step problems. o Make sure units are compatible when creating, simplifying/evaluating, and solving equations. Appropriate units or quantities need to be used when answering realworld situations. o Use labels to put the answers into proper context. Working with expressions, equations, relations and functions can be facilitated by understanding the quantities and their relationships. Graphs should be set up with the appropriate scales and units for the given context. Level of accuracy is dependent on the limitations of measurement within the context of the real-world problem. HS.N-Q.A.1 HS.N-Q.A.1 Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the Mathematical Practices HS.MP.4. Model with mathematics. Essential Questions How can you convert a given quantity in a unit rate to a different unit rate? For example, how can you convert feet per second to miles per hour? Why would you want to be able to convert quantities to different units? How can units and unit relationships be used to set up and solve multi-step problems? Give an example of a real-world situation and explain what unit or quantity you expressed the answer in and why. How can you determine which scale and unit to use when creating a graph to represent a set of data? Examples & Explanations Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions. Include word problems where quantities are given in different units, which must be converted to make sense of the problem. HS.MP.5. Use appropriate tools (Continued on next page) 16 origin in graphs and data displays. strategically. HS.MP.6. Attend to precision. Example: A problem might have one object moving 12 feet per second and another at 5 miles per hour. To compare speeds, students convert 12 feet per second to miles per hour: 12 ft 1 mile 60sec 60min 43200 mile · · · = » 8.18 miles per hour. 1 sec 5280 ft 1 min 1 hr 5280 hr HS.N-Q.A.2 HS.N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling. HS.N-Q.A.3 HS.N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. Mathematical Practices HS.MP.4. Model with mathematics. Graphical representations and data displays include, but are not limited to: line graphs, circle graphs, histograms, multi-line graphs, scatter plots, and multi-bar graphs. Examples & Explanations Examples: What type of measurements would you use to determine your income and expenses for one month? How could you express the number of accidents in Arizona? HS.MP.6. Attend to precision. Mathematic al Practices HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. Examples & Explanations The margin of error and tolerance limit varies according to the measure, tool used, and context. Example: Determining price of gas by estimating to the nearest cent is appropriate because you will not $ 3 . 479 pay in fractions of a cent, but the cost of gas is given to tenths of a cent, e.g., . gallon Additional Domain Information – Quantities (N-Q) Key Vocabulary Unit Descriptive model Unit rate Ratio 17 Scale Equivalent Origin Unit Conversion Example Resources Books Textbook Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Technology http://www.wolframalpha.com/examples/Math.html Useful for checking correct conversions. http://nlvm.usu.edu/en/nav/frames_asid_272_g_4_t_4.html?open=instructions&from=search.html?qt=unit+conversion Useful site with virtual practice problems. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons http://oakroadsystems.com/math/convert.htm#Really1 A nicely thorough introduction to the basics of unit conversion, with practice problems at the end. http://www.virtualnerd.com/pre-algebra/ratios-proportions/rates-word-problem-solution.php The video lesson describes how to use unit conversion to solve word problems, labeling each step of the process carefully. http://www.khanacademy.org/math/arithmetic/basic-ratios-proportions/v/unit-conversion This video lesson on unit conversion explains in detail how metric unit breakdown is used to arrive at different units of the same quantity, and makes a great cross-curricular connection with science. Common Student Misconceptions Students often have difficulty understanding how ratios expressed in different units can be equal to one. For example, and it is permissible to multiply by that ratio. 5280 ft is simply one, 1 mile Students need to make sure to put the quantities in the numerator or denominator so that the terms can cancel appropriately. Example: Convert 140 ft. to miles. In this case they often put 5280 ft in the numerator rather than in the denominator. Students often do not understand that the scale on a graph must be marked in equal intervals. For example, if a table gives the values 1, 3, 4, 9, then students will label constant intervals on their axis with 1, 3, 4, 9, rather than 1 through 9. 18 Assessment Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments, daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above. PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills, concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015. 19 High School Algebra 1 Conceptual Category: Algebra (4 Domains, 8 Clusters) Domain: Seeing Structure in Expressions (2 Clusters) Seeing Structure in Expressions (A-SSE) ) (Domain 1 - Cluster 1 - Standards 1 and 2) Interpret the structure of expressions. (Linear, exponential and quadratic) Essential Concepts Expressions consist of terms (parts being added or subtracted). Terms can either be a constant, a variable with a coefficient, or a coefficient times a variable raised to a power. Real-world problems with changing quantities can be represented by expressions with variables. The relationship between the abstract symbolic representations of expressions can be identified based on how they relate to the given situation. Complicated expressions can be interpreted by viewing parts of the expression as single entities. Structure within an expression can be identified and used to factor or simplify the expression. HS.A-SSE.A.1 HS.A-SSE.A.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. Connection: 9-10.RST.4 b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, n interpret P(1+r) as the product of P and a factor not depending on P. Mathematical Practices HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.7. Look Essential Questions Give an example of a real-world problem and write an expression to model the relationship, and explain how the algebraic symbols represent the words in the problem. How are coefficients and factors related to each other? How does viewing a complicated expression by its single parts help to interpret and solve problems? What does it mean to call something a quantity? How does using the structure of an expression help to simplify the expression? Why would you want to simplify an expression? Examples & Explanations A-SSE.1 starts by being limited to linear expressions and to exponential expressions with integer exponents. Later in the year, focus on quadratic and exponential expressions. A-SSE.1b starts with exponents that are integers and then extends from the integer exponents to rational exponents, focusing on those that represent square or cube roots. Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret their meanings in terms of a context. Examples: n Interpret P(1+r) as the product of P and a factor not depending on P. ) Suppose the cost of cell phone service for a month is represented by the expression 0.40s + 12.95. Students can analyze how the coefficient of 0.40 represents the cost of one minute (40¢), while the constant of 12.95 represents a fixed, monthly fee, and s stands for the number of cell phone minutes used in the month. Similar real-world examples, such as tax rates, can also be used to explore the meaning of expressions. 20 for and make use of structure. HS.A-SSE.A.2 HS.A-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see 4 4 2 2 2 2 x – y as (x ) – (y ) , thus recognizing it as a difference of squares that can be factored as 2 2 2 2 (x – y )(x + y ). Mathematic al Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure. Factor 3x(x – 5) + 2(x – 5). Solution: The “x – 5” is common to both expressions being added, so it can be factored out by the distributive property. The factorization is (3x + 2)(x – 5). Examples & Explanations Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If the remaining expression is quadratic, students should factor the expression further. Example: Factor x 3 - 2x 2 - 35x . 4 4 2 2 2 2 See x – y as (x ) – (y ) , thus recognizing it as a difference of squares that can be factored 2 2 2 2 as (x – y )(x + y ). Note that the first factor can be factored further. Seeing Structure in Expressions (A-SSE) (Domain 1 - Cluster 2 - Standard 3) Write expressions in equivalent forms to solve problems. (Quadratic and exponential) Essential Concepts The solutions of quadratic equations are the x-intercepts of the parabola or zeros of quadratic functions. Factoring methods and the method of completing the square reveal attributes of the graphs of quadratic functions. Factoring a quadratic reveals the zeros of the function. Completing the square in a quadratic equation reveals the maximum or minimum value of the function. Properties of exponents are used to transform expressions for exponential functions. HS.A-SSE.B.3 HS.A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and Mathematical Practices Essential Questions What are the solutions to a quadratic equation and how do they relate to the graph? What attributes of the graph will factoring and completing the square reveal about a quadratic function? How are properties of exponents used to transform expressions for exponential functions? Why would you want to transform an expression for an exponential function? Examples & Explanations It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes handin-hand with understanding what different forms of a quadratic expression reveal. 21 explain properties of the quantity represented by the expression. Connections: 9-10.WHST.1c; 11-12.WHST.1c a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. Students will use the properties of operations to create equivalent expressions. Teachers should foster the idea that changing the forms of expressions, such as factoring or completing the square, or transforming expressions from one exponential form to another, are not independent algorithms that are learned for the sake of symbol manipulations. They are processes that are guided by goals (e.g., investigating properties of families of functions and solving contextual problems). 2 HS.MP.4. Model with mathematics. A pair of coordinates (h, k) from the general form f(x) = a(x – h) + k represents the vertex of the parabola, where h represents a horizontal shift and k represents a vertical shift of the parabola y = 2 x from its original position at the origin. A vertex (h, k) is the minimum point of the graph of the quadratic function if a > 0 and is the maximum point of the graph of the quadratic function if a < 0. Understanding an algorithm for completing the square provides a solid foundation for deriving a quadratic formula. HS.MP.7. Look for and make use of structure. Examples: 3 2 Express 2(x – 3x + x – 6) – (x – 3)(x + 4) in factored form and use your answer to say for what values of x the expression is zero. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression t 1.15 can be rewritten as 1/12 12t 12t (1.15 ) ≈ 1.012 to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. t 1/12 12t The expression 1.15 can be rewritten as (1.15 ) ≈ 1.012 equivalent monthly interest rate if the annual rate is 15%. Write the expression below as a constant multiplied by a power of x and use your answer to decide whether the expression gets larger or smaller as x gets larger. Expression Factor Simplify Polynomial to reveal the approximate (2 x 3 )2 (3x 4 ) ( x 2 )3 Additional Domain Information – Seeing Structure in Expressions (A-SSE) Key Vocabulary 12t Term Exponent Greatest Common Factor Binomial 22 Coefficient Base Quadratic Trinomial Vertex Completing the Square Minimum Maximum Example Resources Books Textbook Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra Uncovering Student Thinking in Mathematics Grades 6-12, How Low Can You Go pg 71 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Technology Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad www.Geogebra.org online software to create visuals www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons http://illuminations.nctm.org/LessonDetail.aspx?id=L761. Predicting your financial future. Students use their knowledge of exponents to compute an investment’s worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit card payments and debt. http://illuminations.nctm.org/Lessons/PowerUp/PowerUp-AS-Voltmeter.pdf. Power up. Students explore addition of signed numbers by placing batteries end to end (in the same direction or opposite directions) and observe the sum of the batteries’ voltages. http://www.nctm.org/uploadedFiles/Journals_and_Books/Books/FHSM/RSM-Task/Horseshoes.pdf Students analyze the structure of algebraic expressions and a graph to determine what information each expression readily contributes about the flight of a horseshoe. This task is particularly relevant to students who are studying (or have studied) various quadratic expressions (or functions). The task also illustrates a step in the mathematical modeling process that involves interpreting mathematical results in a real-world context. http://www.geogebra.org/cms/ Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. http://www.uen.org/Lessonplan/preview.cgi?LPid=26843 Students will identify linear and nonlinear relationships in a variety of contexts. Common Student Misconceptions Students will often combine terms that are not like terms. For example, 2 + 3x = 5x or 3x + 2y = 5xy. Students sometimes forget the coefficient of 1 when adding like terms. For example, x + 2x + 3x = 5x rather than 6x. 23 2 Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x rather than 5x. Students will forget to distribute to all terms when multiplying. For example, 6 (2x + 1) = 12x + 1 rather than 12x + 6. 2 Students may not follow the Order of Operations when simplifying expressions. For example, 4x when x = 3 may be incorrectly evaluated as 4•3 2 = 12 = 144, rather than 4•9 = 36. Another common mistake occurs when the distributive property should be used prior to adding/subtracting. For example, 2 + 3( x – 1) incorrectly becomes 5(x – 1) = 5x – 5 instead of 2 + 3(x – 1) = 2 + 3x – 3 = 3x – 1. 2 Students fail to use the property of exponents correctly when using the distributive property. For example, 3x(2x – 1) = 6x – 3x = 3x instead of 2 simplifying as 3x ( 2x – 1) = 6x – 3x. Students fail to understand the structure of expressions. For example, they will write 4x when x = 3 is 43 instead of 4x = 4•x so when x = 3, 4x = 4•3 2 2 2 2 = 12. In addition, students commonly misevaluate –3 = 9 rather than –3 = –9. Students routinely see –3 as the same as (–3) = 9. A method that may 2 2 clear up the misconception is to have students rewrite as –x = –1•x so they know to apply the exponent before the multiplication of –1. Students frequently attempt to “solve” expressions. Many students add “= 0” to an expression they are asked to simplify. Students need to understand the difference between an equation and an expression. Students commonly confuse the properties of exponents, specifically the product of powers property with the power of a power property. For 2 3 5 6 example, students will often simplify (x ) = x instead of x . Students will incorrectly translate expressions that contain a difference of terms. For example, 8 less than 5 times a number is often incorrectly translated as 8 – 5n rather than 5n – 8. 24 Domain: Arithmetic with Polynomials and Rational Expressions (1 Cluster) Arithmetic with Polynomials and Rational Expressions (A-APR) ) (Domain 2 - Cluster 1 – Standard 1) Perform arithmetic operations on polynomials. (Linear and quadratic) Essential Concepts Adding, subtracting and multiplying two polynomials will yield another polynomial, thus making the system of polynomials closed. Addition and subtraction of polynomials is combining like terms. The distributive property proves why you can combine like terms. Multiplication of polynomials is applying the distributive property. Essential Questions Why is the system of polynomials closed under addition, subtraction and multiplication? How is the system of polynomials similar to and different from the system of integers? How does the distributive property show that you can combine like terms? Explain how the distributive property is used to multiply any size polynomials. HS.A-APR.A.1 HS.A-APR.A.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. Mathematical Practices HS.MP.8. Look for regularity in repeated reasoning. Examples & Explanations Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x. Connection: 9-10.RST.4 In arithmetic of polynomials, a central idea is the distributive property, because it is fundamental not only in polynomial multiplication but also in polynomial addition and subtraction. With the distributive property, there is little need to emphasize misleading mnemonics, such as FOIL, which is relevant only when multiplying two binomials, and the procedural reminder to “collect like terms” as a consequence of the distributive property. For example, when adding the polynomials 3x and 2x, the result can be explained with the distributive property as follows: 3x + 2x = (3 + 2)x = 5x. Additional Domain Information – Arithmetic with Polynomials and Rational Expressions (A-APR) Key Vocabulary Expression Simplify Polynomial Distributive Property Exponential Term Exponent Binomial Trinomial Quadratic Coefficient Base Factor Linear Closure Property Example Resources Books Textbook Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra 25 The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Technology Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad www.Geogebra.org online software to create visuals www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons http://illuminations.nctm.org/Lessons/PowerUp/PowerUp-AS-Voltmeter.pdf Power up. Students explore addition of signed numbers by placing batteries end to end (in the same direction or opposite directions) and observe the sum of the batteries’ voltages. http://wpmu.bionicteaching.com/kmspruill/2009/12/06/lesson-plan-adding-and-subtracting-polynomials/ This lesson includes a clip of how math is used in computer graphics in the movies along with a PowerPoint presentation on adding and subtracting polynomials. http://www.discoveryeducation.com/teachers/free-lesson-plans/rational-number-concepts.cfm Common Student Misconceptions Students often forget to distribute the subtraction to terms other than the first one. For example, students will write (4x + 3) – (2x + 1) = 4x + 3 – 2x + 1 = 2x + 4 rather than 4x + 3 – 2x – 1 = 2x + 2. 2 Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x rather than 5x. Students may not distribute the multiplication of polynomials correctly and only multiply like terms. For example, they will write (x + 3)(x – 2) = 2 2 x – 6 rather than x – 2x + 3x – 6. 26 Domain: Creating Equations (1 Cluster) Creating Equations (A-CED) (Domain 3 – Cluster 1 – Standards 1, 2, 3 and 4) Creating equations that describe numbers or relationships. [Linear, quadratic and exponential (integer inputs only); A-CED.3 linear only] Essential Concepts Equations and inequalities can be created to represent and solve realworld and mathematical problems. Relationships between two quantities can be represented through the creation of equations in two variables and graphed on coordinate axes with labels and scales. Solutions are viable or not in different situations depending upon the constraints of the given context. Formulas can be rearranged and solved for a given variable using the same reasoning as solving an equation. HS.A-CED.A.1 HS.A-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear, quadratic and exponential functions. Essential Questions Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. How do you translate real-world situations into mathematical equations and inequalities? How do you determine if a situation is best represented by an equation, an inequality, a system of equations or a system of inequalities? Why would you want to create an equation or inequality to represent a real-world problem? How are graphs of equations and inequalities similar and different? How do you determine if a given point is a viable solution to a system of equations or inequalities, both on a graph and using the equations? Why would you want to solve a given formula for a particular variable? How do you solve a given formula for a particular variable? Examples & Explanations Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. Start with work on linear and exponential equations, then, later in the year, extend to quadratic equations. HS.MP.4. Model with mathematics. Equations can represent real-world and mathematical problems. Include equations and inequalities that arise when comparing the values of two different functions, such as one describing linear growth and one describing exponential growth. HS.MP.5. Use appropriate tools strategically. Examples: 2 Given that the following trapezoid has area 54 cm , set up an equation to find the length of the unknown base, and solve the equation. Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by h(t ) 16t 2 64t 936 . After how many seconds does the lava reach its maximum height of 1000 feet? 27 t The value of an investment over time is given by the equation A(t) = 10,000(1.03) . What does each part of the equation represent? Solution: The $10,000 represents the initial value of the investment. The 1.03 means that the investment will grow exponentially at a rate of 3% per year for t years. You bought a car at a cost of $20,000. Each year that you own the car the value of the car will decrease at a rate of 25%. Write an equation that can be used to find the value of the car after t years. t Solution: C(t) = $20,000(0.75) . The base is 1 – 0.25 = 0.75 and is between 0 and 1, representing exponential decay. The value of $20,000 represents the initial cost of the car. An amount of $100 was deposited in a savings account on January 1st in each of the years 2010, 2011, 2012, and so on to 2020, with an annual yield of 7%. What will be the balance in the savings account at the end of the day on January 1, 2020? In your solution, illustrate the use of a formula for a geometric series when S n represents the value of the geometric series with the first term g, constant ratio r ≠ 1, and n + 1 terms. Before using the formula, it might be reasonable to demonstrate the way the formula is derived. Solution: Sn Multiply by r: rSn Subtract: Sn – rSn Factor: Sn(1 – r) Divide by (1 – r): Sn = = = = = 2 3 n g + gr + gr + gr + … + gr 2 3 n n+1 gr + gr + gr + … + gr + gr n+1 g – gr n+1 g(1 – r ) n+1 g(1 – r )/(1 – r) 10 HS.A-CED.A.2 HS.A-CED.A.2 Create equations in two or more variables to represent relationships between The amount of the investment on January 1, 2020 can be found using: 100(1.07) + 9 100(1.07) + … + 100(1.07) + 100. If the first term of this geometric series is g = 100, the ratio is 1.07, and n = 10, the formula for the value of the geometric series gives S 10 = $1578.36 to the nearest cent. Mathematical Practices HS.MP.2. Reason Examples & Explanations Limit A-CED.1 and A-CED.2 to linear and exponential equations, and, in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs. 28 quantities; graph equations on coordinate axes with labels and scales. HS.A-CED.A.3 HS.A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. HS.A-CED.A.4 HS.A-CED.A.4 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. abstractly and quantitatively. Start with work on linear and exponential equations, then, later in the year, extend to quadratic equations. HS.MP.4. Model with mathematics. Example: 2 The formula for the surface area of a cylinder is given by V = πr h, where r represents the radius of the circular cross-section of the cylinder and h represents the height. Choose a fixed value for h and graph V vs. r. Then pick a fixed value for r and graph V vs. h. Compare the graphs. What is the appropriate domain for r and h? Be sure to label your graphs and use an appropriate scale. HS.MP.5. Use appropriate tools strategically. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. Examples & Explanations Limit A-CED.3 to linear equations and inequalities. Examples: A club is selling hats and jackets as a fundraiser. Their budget is $1500 and they want to order at least 250 items. They must buy at least as many hats as they buy jackets. Each hat costs $5 and each jacket costs $8. o Write a system of inequalities to represent the situation. o Graph the inequalities. o If the club buys 150 hats and 100 jackets, will the conditions be satisfied? o What is the maximum number of jackets they can buy and still meet the conditions? Represent inequalities describing nutritional and cost constraints on combinations of different foods. Examples & Explanations Start by limiting A-CED.4 to formulas that are linear in the variable of interest. Later in the year, extend to formulas involving squared variables. Examples: The Pythagorean theorem expresses the relation between the legs a and b of a right 2 2 2 triangle and its hypotenuse c with the equation a + b = c . o Why might the theorem need to be solved for c? o Solve the equation for c and write a problem situation where this form of the equation might be useful. Solve V = 4 3 p r for radius r. 3 29 HS.MP.7. Look for and make use of structure. Motion can be described by the formula below, where t = time elapsed, u = initial velocity, a = acceleration, and s = distance traveled. 2 s = ut+½at o o Why might the equation need to be rewritten in terms of a? Rewrite the equation in terms of a. Rearrange Ohm’s law V = IR to highlight resistance R. Additional Domain Information – Creating Equations (A-CED) Key Vocabulary Expression Simplify Solution Zeros Graph Example Resources Books Equation Linear Exponent X-intercept System Inequality Quadratic Y-intercept Formula Coordinate axes Textbook Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 2: Building Equations and Functions The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Mission Mathematics II: Grades 9-12, Modeling Space-Debris Accumulation pg 28 Active Algebra: Strategies and Lessons for Successfully Teaching Linear Relationships, Section II: Linear Relations: Lessons and Assessments Technology Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad www.Geogebra.org online software to create visuals www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons 30 www.smartskies.nasa.gov/lineup. LineUp with Math. This simulation is an interactive online simulator featuring air traffic control problems in a realistic route structure with 2 to 5 planes. Teacher guide, student website, and teacher website with materials included on website. http://illuminations.nctm.org/LessonDetail.aspx?ID=L713. Students work collaboratively to come up with a bargaining plan to trick the raja into feeding the village using algebra, exponential growth, and estimation. http://www.uen.org/Lessonplan/preview.cgi?LPid=19825 Students go car shopping online, investigate the relationship between variables such as interest rate and monthly payment, develop two payment plans using online loan calculators, write a slope-intercept equation for each plan, and create a graph and table for the equations using a graphing calculator. Common Student Misconceptions Students may interchange slope and y-intercept when creating equations. For example, a taxi cab costs $4 for a dropped flag and charges $2 per mile. Students may fail to see that $2 is a rate of change and is slope while the $4 is the starting cost and incorrectly write the equation as y = 4x + 2 instead of y = 2x + 4. Given a graph of a line, students use the x-intercept for b instead of the y-intercept. Given a graph, students incorrectly compute slope as run over rise rather than rise over run. For example, they will compute slope with the change in x over the change in y. Students do not know when to include the “or equal to” bar when translating the graph of an inequality. Students do not correctly identify whether a situation should be represented by a linear, quadratic, or exponential function. Students often do not understand what the variables represent. For example, if the height h in feet of a piece of lava t seconds after it is ejected 2 from a volcano is given by h(t) = -16t + 64t + 936 and the student is asked to find the time it takes for the piece of lava to hit the ground, the student will have difficulties understanding that h = 0 at the ground and that they need to solve for t. Students have difficulties rearranging formulas to highlight a different quantity. For example, many students will not see that solving 5x = 10 by dividing both sides by 5 is the same as solving for b in the equation ab = c by dividing both sides by a. 31 Domain: Reasoning with Equations and Inequalities (4 Clusters) Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 1 – Standard 1) Understand solving equations as a process of reasoning and explain the reasoning (Master linear; learn as general principle) Essential Concepts Equations are solved as a process of reasoning using properties of operations and equality, which can justify each step of the process. A solution to an equation can be checked, by substituting in that value for the variable and simplifying to see if the equation holds true. HS.A-REI.A.1 HS.A-REI.A.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. Construct a viable argument to justify a solution method. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.7. Look for and make use of structure. Essential Questions What do you use to justify your reasoning when solving an equation? How do you determine if an equation is solved properly? How do you determine and justify if a solution to an equation is correct? Why are properties of real numbers important when solving equations? Examples & Explanations Students should focus on and master A-REI.1 for linear equations and be able to extend and apply their reasoning to other types of equations for later in the year and in future courses. Students will solve exponential equations with logarithms in Algebra II. Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions. Other operations, such as squaring both sides, may produce equations that have extraneous solutions. Each step of solving an equation can be defended, much like providing evidence for steps of a geometric proof. Provide examples for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation; the order of steps taken will not matter. Examples: Explain why the equation x/2 + 7/3 = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean that x/2 + 7/3 is equal to 3x + 14? Show that x = 2 and x = –3 are solutions to the equation x 2 x 6 . Write the equation in a form that shows these are the only solutions, explaining each step in your reasoning. Transform 2x – 5 = 7 to 2x = 12 and tell what property of equality was used. 2x 5 7 Solution: 2 x 5 5 7 5 Additionproperty of equality. 2 x 12 32 Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 2 – Standards 3 and 4) Solve equations and inequalities in one variable. Linear inequalities; linear equations with letter coefficients; quadratics with real solutions Essential Concepts Equations and inequalities are solved using properties of operations, equality, and inequality, which can justify each step of the process. A solution to an equation can be checked, by substituting in that value for the variable and simplifying to see if the equation or inequality holds true. Laws of exponents can be used to solve simple exponential equations. Completing the square can be used to transform a quadratic equation 2 into the form (x-p) = q. 2 The quadratic formula can be derived by completing the square of ax + bx+c = 0. Quadratic equations can be solved by a variety of methods, for example: by inspection, graphing, taking square roots, factoring, completing the square and the quadratic formula. Quadratic equations can have extraneous and/or complex solutions. HS.A-REI.B.3 HS.A-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Mathematical Practices HS.MP.2 Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning. Essential Questions What do you use to justify your steps when solving linear and nonlinear equations and inequalities? How do you determine and justify whether a solution to an equation or inequality is correct? How do operations performed on real numbers affect the relationship between the quantities in an inequality? Why would you want to transform a quadratic equation to the form 2 (x-p) = q? How do you determine which method is best for solving a quadratic equation? Why do some quadratic equations have extraneous and/or complex solutions? Examples & Explanations Extend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple exponential x x equations that rely only on application of the laws of exponents, such as 5 =125 or 2 =1/16. Examples: Solve for the variable: 7 - y - 8 = 111 3 3x > 9 ax + 7 = 12, when a = 2 3+ x x -9 = 7 4 Solve for x: 2/3x + 9 < 18 33 HS.A-REI.B.4 HS.A-REI.B.4 Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – 2 p) = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection 2 (e.g., for x = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning. Examples & Explanations Students should solve by factoring, completing the square, and using the quadratic formula. The zero product property is used to explain why the factors are set equal to zero. Students should relate the value of the discriminant to the type of root to expect. A natural extension would be to 2 2 relate the type of solutions to ax + bx + c = 0 to the behavior of the graph of y = ax + bx + c. Value of Discriminant 2 b – 4ac = 0 2 b – 4ac > 0 2 b – 4ac < 0 Nature of Roots 1 real root 2 real roots 2 complex roots Nature of Graph intersects x-axis once intersects x-axis twice does not intersect x-axis Examples: 2 Are the roots of 2x + 5 = 2x real or complex? How many roots does it have? Find all solutions of the equation. 2 What is the nature of the roots of x + 6x + 10 = 0? Solve the equation using the quadratic formula and completing the square. How are the two methods related? Projectile motion problems, in which the initial conditions establish one of the solutions as extraneous within the context of the problem. o An object is launched at 14.7 meters per second (m/s) from a 49-meter tall platform. 2 The equation for the object's height s at time t seconds after launch is s(t) = –4.9t + 14.7t + 49, where s is in meters. When does the object strike the ground? 0 = -4.9t 2 +14.7t + 49 2 Solution: 0 = t - 3t -10 0 = (t + 2)(t - 5) So the solutions for t are t = 5 or t = –2, but t = –2 does not make sense in the context of this problem and therefore is an extraneous solution. Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II. In Algebra 1, student should be able to recognize when the solution to a quadratic equation yields a complex solution; however writing the solution in the complex form a ± bi for real numbers a and b will be addressed in Algebra II. 34 Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 3 – Standards 5, 6 and 7) Solve systems of equations (Linear-linear and linear-quadratic) Essential Concepts A system of linear equations can have one solution, infinitely many solutions, or no solution. A system of linear equations can be solved graphically, algebraically using elimination/linear combination, substitution, or modeling. Solving a system of equations algebraically yields an exact solution; solving by graphing yields an approximate solution. Multiplying both sides of an equation by a non-zero constant does not change the solution to the equation. Elimination/linear combination is a method of solving a system of linear equations in which the equations are added together in order to eliminate a variable. In elimination/linear combination you may need to multiply one or both of the equations by a non-zero constant in order to be able to eliminate one of the variables. Substitution is a method of solving a system of equations where one equation is solved for a variable and then that expression is substituted into the other equation for that variable, in order to eliminate that variable. A system of a linear equation and a quadratic equation can be solved algebraically using substitution or graphically by finding the points of intersection. HS.A-REI.C.5 HS.A-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Essential Questions How do you determine the number of solutions that a system of equations will have? How do you determine the best method for solving a given system of equations? Why would you want to multiply an equation by a constant (that is not zero)? Why does graphing a system of equations yield an approximate solution as opposed to an exact solution? How can you prove that no matter which method you choose to solve a system of equations, you will always get the same solution? Mathematical Practices Examples & Explanations HS.MP.3. Construct viable arguments and critique the reasoning of others. Systems of linear equations can also have one solution, infinitely many solutions or no solutions. Students will discover these cases as they graph systems of linear equations and solve them algebraically. HS.MP.2. Reason abstractly and quantitatively. Build on student experiences in graphing and solving systems of linear equations from middle school to focus on justification of the methods used. Include cases where the two equations describe the same line (yielding infinitely many solutions) and cases where two equations describe parallel lines (yielding no solution); connect to GPE.5 when it is taught in Geometry, which requires students to prove the slope criteria for parallel lines. A system of linear equations whose graphs meet at one point (intersecting lines) has only one solution, the ordered pair representing the point of intersection. A system of linear equations whose graphs do not meet (parallel lines) has no solutions and the slopes of these lines are the same. A 35 system of linear equations whose graphs are coincident (the same line) has infinitely many solutions, the set of ordered pairs representing all the points on the line. By making connections between algebraic and graphical solutions and the context of the system of linear equations, students are able to make sense of their solutions. Students need opportunities to work with equations and context that include whole number and/or decimals/fractions. Examples: Find x and y using elimination and then using substitution. 3x + 4y = 7 –2x + 8y = 10 Given that the sum of two numbers is 10 and their difference is 4, what are the numbers? Explain how your answer can be deduced from the fact that the two numbers, x and y, satisfy the equations x + y = 10 and x – y = 4. HS.A-REI.C.6 HS.A-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. Connection: ETHS-S6C2-03 Mathematical Practices Examples & Explanations HS.MP.4. Model with mathematics. Examples: José had 4 times as many trading cards as Phillipe. After José gave away 50 cards to his little brother and Phillipe gave 5 cards to his friend for his birthday, they each had an equal amount of cards. Write a system to describe the situation and solve the system. HS.MP.2. Reason abstractly and quantitatively. HS.MP.5. Use appropriate tools strategically. The system solution methods can include but are not limited to graphical, elimination/linear combination, substitution, and modeling. Systems can be written algebraically or can be represented in context. Students may use graphing calculators, programs, or applets to model and find approximate solutions for systems of equations. HS.MP.6. Attend to precision. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in Solve the system of equations: x+ y = 11 and 3x – y = 5. Use a second method to check your answer. Solve the system of equations: x – 2y + 3z = 5, x + 3z = 11, 5y – 6z = 9. 36 repeated reasoning. The opera theater contains 1,200 seats, with three different prices. The seats cost $45 per seat, $50 per seat, and $60 per seat. The opera needs to gross $63,750 on seat sales. There are twice as many $60 seats as $45 seats. How many seats at each price need to be sold? Reasoning with Equations and Inequalities (A-REI) (Domain 4 – Cluster 4 – Standards 10, 11 and 12) Represent and solve equations and inequalities graphically. (Linear and exponential; learn as general principle) Essential Concepts The graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane. Solving a system of equations algebraically yields an exact solution; solving by graphing or by comparing tables of values yields an approximate solution. The solutions (solution set) of a linear inequality in two variables are represented graphically as a half-plane. The solution set of a system of linear inequalities in two variables is the intersection of the corresponding half-planes. HS.A-REI.D.10 HS.A-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.A-REI.D.11 HS.A-REI.D.11 Explain why the x-coordinates of the points where the Mathematical Practices HS.MP.2. Essential Questions How do you determine if a given ordered pair is a solution to an equation? Why are the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect equal to the solutions of the equation f(x) = g(x)? (Continued on next page) When graphing a linear inequality, how do you determine which halfplane to shade in order to represent the solution set? How do you represent the solution set of a system of linear inequalities on a graph? Examples & Explanations For A-REI.10, focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. Examples: Which of the following points would be on the graph of the equation 3x+4y=24 ? (a) (0, 6) (b) (-1, 7) (c) (4/3, 5) (d) (3, 4) Graph the equation and determine which of the following points are on the graph of x y = 3 + 1. (a) (2, 7) (b) (-1, 4/3) (c) (2, 10) (d) (0, 1) Examples & Explanations For A-REI.11, focus on cases where f(x) and g(x) are linear, quadratic and exponential. In Algebra II, students will extend this standard to include higher-order polynomials, rational, absolute value 37 graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. HS.A-REI.D.12 HS.A-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Reason abstractly and quantitatively. and logarithmic functions. HS.MP.4. Model with mathematics. Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions. HS.MP.5. Use appropriate tools strategically. Example: Given the following equations, determine the x value that results in an equal output for both functions. HS.MP.6. Attend to precision. Mathematical Practices HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. f (x ) = 3x - 2 g ( x ) = ( x + 3)2 - 1 Examples & Explanations Students may use graphing calculators, programs, or applets to model and find solutions for inequalities or systems of inequalities. Examples: Graph the solutions: y < 2x + 3. A publishing company publishes a total of no more than 100 magazines every year. At least 30 of these are women’s magazines, but the company always publishes at least as many women’s magazines as men’s magazines. Find a system of inequalities that describes the possible number of men’s and women’s magazines that the company can produce each year consistent with these policies. Graph the solution set. Graph the system of linear inequalities below and determine if (3, 2) is a solution to the system. ìx - 3y > 0 ï íx + y £ 2 ï x + 3 y > -3 î Solution: 38 (3, 2) is not an element of the solution set (graphically or by substitution). Additional Domain Information – Reasoning with Equations and Inequalities (A-REI) Key Vocabulary Equation Inequality Quadratic formula Extraneous solutions Point of Intersection Half-Plane Solution Laws of Exponents Square roots Complex solutions Discriminant Radical Substitution Completing the Square Factoring Elimination/Linear Combinations System Example Resources Books Textbook Focus in High School Mathematics: Reasoning and Sense Making, Chapter 2: Building Equations and Functions The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Technology Key Curriculum Press, Exploring Algebra I with the Geometer’s Sketchpad www.Geogebra.org online software to create visuals www.Khanacademy.org Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons http://www.uen.org/Lessonplan/preview.cgi?LPid=19825 Students go car shopping online, investigate the relationship between variables such as interest rate and monthly payment, develop two payment plans using online loan calculators, write a slope-intercept equation for each plan, and create a graph and table for the equations using a graphing calculator. http://www.uen.org/Lessonplan/preview.cgi?LPid=20241 Students watch a short role play in which a doctor performs an operation on a patient without first diagnosing his condition. They relate this medical malpractice to a mathematician performing an operation before evaluating conditions or deciding what outcome he wants. Next they practice 'diagnosing' x's condition and performing the correct inverse operation to get x by itself. http://www.uen.org/Lessonplan/preview.cgi?LPid=23514 Write and solve simple inequalities. 39 Common Student Misconceptions Students often forget to flip the direction of the inequality sign when multiplying or dividing by a negative number. Students commonly believe that there is only one correct method to solve an equation or inequality. Students often perform inverse operations on the same side of the equation, including combining non-like terms. For example, 3x + 2 = 8 instead of 3x + 2 = 8 -2 -2 -2 -2 x=8 3x = 6 Students will often drop the negative sign at the end or just stop solving for x ,if the coefficient of x is –1. For example, 3x + 2 – 4x = 5 –x + 2 = 5 –x = 3, so a student will say that the solution is 3 rather than –3. Students will be confused as to when there is no real number solution or when the solution set is all real numbers for an equation or system of equations. For example, students will be confused when they try to solve an equation and end up with –2 = 9, which means there are no solutions, or when the solution to a system of equations gives x = x, which means the solution set is all real numbers. 2 Students fail to see that there may still be real number solutions even though a quadratic doesn’t factor. For example, x + x – 4 = 0 does not factor, but students should use the quadratic formula to find the exact solutions. Students will substitute incorrectly in a system. There is a variety of ways that students may incorrectly solve for a variable to be substituted in to the other equation in a system. In addition, students may erroneously substitute back into the same equation and believe the solution set is infinite. 2 2 Students often fail to give both solutions when solving x = b type equations. For example, given x = 49, students often only state x = 7 rather than x = ± 7. Students will believe that the variable can only be on the left side of the equation. Assessment Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments (e.g., pre-assessments, daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above. PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills, concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015. 40 High School Algebra 1 Conceptual Category: Functions (3 Domains, 7 Clusters) Domain: Interpreting Functions (3 Clusters) Interpreting Functions (F-IF) (Domain 1 – Cluster 1 – Standards 1, 2 and 3) Understand the concept of a function and use function notation. (Learn as a general principle; focus on linear and exponential and on arithmetic and geometric sequences.) Essential Concepts A function is a rule that assigns each input exactly one output. In function notation, f(x) denotes that f is a function of x. The set of all inputs (x) for a function is called the domain; the set of all outputs (f(x)) for a function is called the range. The domain and range of a function can be expressed as a set of numbers using set notation, an inequality, or as a graphed solution. The graph of a function f is the graph of the equation y = f(x). Algebraic equations, written in function notation, can be used to evaluate functions for a given input. For a function f(x), f(a) represents the value of the function when x = a. Sequences are functions that have a discrete domain, which is a subset of the integers. A recursive sequence is a sequence in which each term is built upon the previous term. HS.F-IF.A.1 HS.F-IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Mathematical Practices HS.MP.2 Reason abstractly and quantitatively. Essential Questions Given a table or graph, how do you determine if it represents a function? How is a graph related to its algebraic function? How could you use function notation to represent a specific output of a function? How can the Fibonacci sequence be used to explain a recursive pattern? How can you describe a sequence as a function? Examples & Explanations Students should experience a variety of types of situations modeled by functions. Detailed analysis of any particular class of functions at this stage is not advised. Students should apply these concepts throughout their future mathematics courses. Draw examples from linear, quadratic, and exponential functions. The domain is the set of all inputs (x values); the range is the set of all outputs (y = f(x)). The domain of a function may be given by an algebraic expression. Unless otherwise specified, it is the largest possible domain. 2 For example, the rule that takes x as input and gives x +5x+4 as output is a function. Using y to 2 stand for the output we can represent this function with the equation y = x +5x+4, and the graph of the equation is the graph of the function. Students are expected to use function notation such as 2 f(x) = x +5x+4. 41 Example: Determine which of the following tables represent a function and explain why. A B x f(x) x f(x) 0 0 0 1 1 2 1 2 1 3 2 2 4 5 3 4 Solution: A represents a function because for each element in the domain there is exactly one element in the range. B does NOT represent a function because when x = 1, there are two values for f(x): 2 and 3. HS.F-IF.A.2 HS.F-IF.A.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. Connection: 9-10.RST.4 HS.F-IF.A.3 HS.F-IF.A.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. Mathematical Practices HS.MP.8. Look for and express regularity in repeated reasoning. Examples & Explanations Examples: If f ( x ) = x 2 + 4x - 12 , find f (2). 1 ) , f (a ) , and f (a - h). 2 If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000 and P(10)-P(9) = 5,900. Let f ( x ) = 2( x + 3)2 . Find f (3) , f (- Examples & Explanations In F-IF.3, draw a connection to F.BF.2, which requires students to write arithmetic and geometric sequences. Emphasize arithmetic and geometric sequences as examples of linear and exponential functions. Example: The Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. 42 Interpreting Functions (F-IF) (Domain 1 – Cluster 2 – Standards 4, 5 and 6) Interpret functions that arise in applications in terms of a context. (Linear, exponential, and quadratic) Essential Concepts Key features of a graph or table may include intercepts, intervals in which the function is increasing, decreasing or constant, intervals in which the function is positive, negative or zero, symmetry, maxima, minima, and end behavior. Given a verbal description of a relationship that can be modeled by a function, a table or graph can be constructed and used to interpret key features of that function. The meaning of the key features of a graph or table, such as domain, range, rate of change and intercepts, can be interpreted in the context of a problem. The intervals over which a function is increasing, decreasing or constant, positive, negative or zero are subsets of the function’s domain. The appropriate domain for a function describing a real-life situation may be smaller than the largest possible domain. The average rate of change of a function y = f(x) over an interval [a,b] is Dy f (b) - f (a) . = Dx b-a HS.F-IF.B.4 HS.F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maxima and minima; symmetries; end Essential Questions How can you describe the shape of a graph? How can you relate the shape of a graph to the meaning of the relationship it represents? How would you determine the appropriate domain for a function describing a real-life situation? Given a function that describes a real-life situation, what can the average rate of change of the function tell you? How do the parts of a graph of a function relate to its real world context? Mathematical Practices Examples & Explanations HS.MP.2. Reason abstractly and quantitatively. Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on quadratic functions and compare them with linear and exponential functions. In Algebra II, students will extend this standard to include higher order polynomials, rational, absolute value, and trigonometric functions. HS.MP.4. Model with mathematics. Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. HS.MP.5. Use appropriate tools strategically. Examples: A rocket is launched from 180 feet above the ground at time t = 0. The function that models 2 this situation is given by h = – 16t + 96t + 180, where t is measured in seconds and h is height above the ground measured in feet. o What is a reasonable domain restriction for t in this context? o Determine the height of the rocket two seconds after it was launched. HS.MP.6. Attend 43 behavior; and periodicity. to precision. o o o o Connections: ETHS-S6C2.03; 9-10.RST.7; 11-12.RST.7 HS.F-IF.B.5 HS.F-IF.B.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. HS.F-IF.B.6 HS.F-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. Determine the maximum height obtained by the rocket. Determine the time when the rocket is 100 feet above the ground. Determine the time at which the rocket hits the ground. How would you refine your answer to the first question based on your response to the second and fifth questions? 2 x Compare the graphs of y = 3x and y = 3 . Let f (x) = -x 2 - 5x +1. Graph the function and identify end behavior and any intervals of constancy, increase, and decrease. It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total rainfall of 3 inches. It didn’t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a function of time, from midnight to midday. Examples & Explanations Start F-IF.4 and 5 by focusing on linear and exponential functions. Later in the year, focus on quadratic functions and compare them with linear and exponential functions. Students may explain the existing relationships orally or in written format. Example: If the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. HS.MP.6. Attend to precision. Mathematic al Practices HS.MP.2. Reason abstractly and quantitatively. Examples & Explanations Start F-IF.6 by focusing on linear functions and exponential functions whose domain is a subset of the integers. Later in the year, focus on quadratic functions and compare them with linear and exponential functions. The Algebra II course will address other types of functions. The average rate of change of a function y = f(x) over an interval [a,b] is Dy f (b) - f (a) . In = Dx b-a addition to finding average rates of change from functions given symbolically, graphically, or in a table, 44 Connections: ETHS-S1C2-01; 9-10.RST.3 HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. students may collect data from experiments or simulations (ex. falling ball, velocity of a car, etc.) and find average rates of change for the function modeling the situation. Examples: Use the following table to find the average rate of change of g over the intervals [-2, -1] and [0,2]: x -2 -1 0 2 g(x) 2 -1 -4 -10 The table below shows the elapsed time when two different cars pass a 10, 20, 30, 40 and 50 meter mark on a test track. o For car 1, what is the average velocity (change in distance divided by change in time) between the 0 and 10 meter mark? Between the 0 and 50 meter mark? Between the 20 and 30 meter mark? Analyze the data to describe the motion of car 1. o How does the velocity of car 1 compare to that of car 2? d 10 20 30 40 50 Car 1 T 4.472 6.325 7.746 8.944 10 Car 2 t 1.742 2.899 3.831 4.633 5.348 Interpreting Functions (F-IF) (Domain 1 – Cluster 3 – Standards 7, 8 and 9) Analyze functions using different representations. (Linear, exponential, quadratic, absolute value, step and piecewise-defined) Essential Concepts To graph a function you can create a table of values, analyze the equation, or use a graphing calculator. Key features of a graph or table may include intercepts, intervals in which the function is increasing, decreasing or constant, intervals in which the function is positive, negative or zero, symmetry, maxima, minima, and end behavior. A linear function can be written in point-slope, slope-intercept or Essential Questions 45 How can you compare properties of two functions if they are represented in different ways? How can you determine which form of a function is best for a given situation? Why might you need to complete the square? How could you determine if a function represents exponential growth or exponential decay? standard form. A quadratic function can be written in vertex or standard form. Factoring a quadratic function will help to determine the zeros. Completing the square will help determine the vertex of the graph. For a function of the form f (t) = a ×bt , if b>1 the function represents exponential growth; if b<1 the function represents exponential decay. HS.F-IF.C.7 HS.F-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Mathematical Practices HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. Examples & Explanations For F-IF.7a, 7e, focus only on linear and exponential functions. Include comparisons of two functions presented algebraically. For example, compare the growth of two linear functions, or two n n exponential functions such as y=3 and y=100*2 . For F-IF.7b, compare and contrast absolute value, step and piecewise-defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise-defined functions. For F-IF.7e focus only on exponential functions. Key characteristics include but are not limited to maxima, minima, intercepts, symmetry, end behavior, and asymptotes. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Connections: ETHS-S6C1-03; ETHS-S6C2-03 Examples: Graph the function f(x) = │x – 3│ + 5 and describe key characteristics of the graph b. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. x Graph the function f(x) = 2 by creating a table of values. Identify the key characteristics of the graph. Connections: ETHS-S6C1-03; ETHS-S6C2-03 HS.F-IF.C.8 HS.F-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Mathematical Practices HS.MP.2. Reason abstractly and Examples & Explanations F-IF.8a will be taught later in the year, because it focuses on quadratic factoring and completing the square. In Algebra II students will extend their work on F-IF.8 to focus on applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. 46 Connection: 11-12.RST.7 a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. quantitatively. HS.MP.7. Look for and make use of structure. Example: 2 Factor the following quadratic to identify its zeros: x + 2x - 8 = 0 Mathematical Practices Examples & Explanations 2 Complete the square for the quadratic and identify its vertex: x + 6x +19 = 0 Connection: 11-12.RST.7 HS.F-IF.C.9 HS.F-IF.C.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Connections: ETHS-S6C1-03; ETHS-S6C2-03; 9-10.RST.7 HS.MP.6. Attend to precision. HS.MP.7. Look for and make use of structure. Start F-IF.9 by focusing on linear and exponential functions. Include comparisons of two functions presented algebraically. Later in the year focus on expanding the types of functions to include linear, exponential, and quadratic. Extend work with quadratics to include the relationship between coefficients and roots, and once roots are known, a quadratic equation can be factored. Example: Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Examine the functions below. Which function has the larger maximum? How do you know? f (x) = -2x 2 -8x + 20 47 Additional Domain Information – Interpreting Functions (F-IF) Key Vocabulary Average Rate of Change Function Input/Domain Output/Range Linear Function Piecewise Function Quadratic Function Exponential Function Trigonometric Function Period Midline Amplitude Sequence Fibonacci Sequence Recursive Arithmetic Sequence Geometric Sequence Maxima Minima End Behavior Point-Slope Form Slope-Intercept Form Standard Form Vertex Form x-intercept y-intercept Example Resources Books Developing Essential Understanding of Functions: Grades 9-12 by NCTM The X’s and Why’s of Algebra: Key Ideas and Common Misconceptions by Anne Collins & Linda Dacey Technology Interpreting functions: http://www.brightstorm.com/math/algebra/graphs-and-functions/interpreting-graphs-problem-1/ Video on interpreting graphs for students in Algebra. Shows students how to look at what graphs mean and how to interpret slopes using real world situations. http://www.khanacademy.org/ Conceptual videos: Linear, Quadratic, Exponential Functions Basic Linear Function, Recognizing Linear Functions, Linear Function Graphs, Graphing a Quadratic Function, Applying Quadratic Functions 2, Quadratic Functions 1, Applying Quadratic Functions 3, Applying Quadratic Functions 1, Quadratic Functions 2, Quadratic Functions 3, Exponential Growth Functions, Exponential Decay Functions, See all videos from topic ck12.org Algebra 1 Examples, Graphing Exponential Functions http://www.schooltube.com/video/f44d3551932486f18d81/ Provides examples on how to build & analyze functions. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. 48 Example Lessons http://www.illustrativemathematics.org/standards/hs Understand the concept of a function and use function notation. Interpret functions that arise in applications in terms of the context. F-IF: Interpret functions that arise in applications in terms of the context. Common Student Misconceptions Students commonly have difficulty understanding the function notation Y = f(x). For example, f(2) = 5 is the function notation for the point (2, 5). Students can have difficulty identifying the x and y values for a given point from the function notation. Students often confuse the domain with the range. Students commonly have difficulty identifying the zeros of a function. Students think that the zero of the function is when x = 0, as opposed to when y = 0. Students misunderstand how to write the interval where a function is increasing or decreasing. Often students write the answer in terms of yinterval rather than in terms of the x-interval. Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example, 2 2 2 to shift the graph of y = x left by 6 units, they may write y = (x - 6) rather than y = (x + 6) . Students believe that the domain of a function is the same, regardless of the contextual problem that it is modeling. For example, the domain of a quadratic includes negative values, but for a quadratic modeling the height of a falling object as a function of time t, the domain should be t ≥ 0. Students often confuse the independent variable with the dependent variable. Students often have difficulty identifying the quantities to be represented by variables, in modeling problems. 49 Domain: Building Functions (2 Clusters) Building Functions (F-BF) (Domain 2 – Cluster 1 – Standards 1 and 2) Build a function that models a relationship between two quantities. (For F-BF.1 and 2, linear, exponential and quadratic) Essential Concepts A function is a relationship between two quantities. The function representing a given situation may be a combination of more than one standard function. Standard functions may be combined through arithmetic operations. Arithmetic and geometric sequences can be written both recursively and with an explicit formula. A recursive formula for a sequence describes how to determine the next term from the previous term(s). An explicit formula for a sequence describes how to determine any term in the sequence. Arithmetic sequences can be described by linear functions. Geometric sequences can be described by exponential functions. Sequences model situations in which the domain is a set of integers. HS.F-BF.A.1 HS.F-BF.A.1 Write a function that describes a relationship between two quantities. Connections: ETHS-S6C1-03; ETHS-S6C2-03 a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Connections: ETHS-S6C1-03; ETHS-S6C2-03; 9-10.RST.7; 11-12.RST.7 Mathematical Practices HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. Essential Questions What data would you need to write a linear, basic quadratic, or basic exponential function? How do you translate a description of the relationship between two quantities into an algebraic equation or inequality? Why are arithmetic sequences described by linear functions? Why are geometric sequences described by exponential functions? How could you translate a recursive formula for a sequence into an explicit formula? Vice versa? Examples & Explanations Start by limiting F-BF.1a, 1b to linear and exponential functions. Later in the year extend this work to focus on situations that exhibit a quadratic relationship. Students will analyze a given problem to determine the function expressed by identifying patterns in the function’s rate of change. They will specify intervals of increase, decrease, constancy, and, if possible, relate them to the function’s description in words or graphically. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model functions. Examples: You buy a $10,000 car with an annual interest rate of 6 percent compounded annually and make monthly payments of $250. Express the amount remaining to be paid off as a function of the number of months, using a recursion equation. A cup of coffee is initially at a temperature of 93º F. The difference between its temperature and the room temperature of 68º F decreases by 9% each minute. Write a function describing the temperature of the coffee as a function of time. HS.MP.6. Attend 50 to precision. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning. Building Functions (F-FB) (Domain 2 – Cluster 2 – Standards 3 and 4) Build new functions from existing functions. (Linear, exponential, quadratic and absolute value; for F-BF.4a, linear only) Essential Concepts f(x) + k will translate the graph of the function f(x) up or down by k units. k f(x) will expand or contract the graph of the function f(x) vertically by a factor of k. If k<0 the graph will reflect across the x-axis. f(kx) will expand or contract the graph of the function f(x) horizontally by a factor of k. If k<0 the graph will reflect across the y-axis. f(x + k) will translate the graph of the function f(x) left or right by k units. If f(-x) = f(x) then the function is even, therefore its graph is symmetrical across the y-axis. If f(-x) = - f(x) then the function is odd, therefore its graph is symmetrical across the origin. Two functions f and g are inverses of one another if for all values of x in the domain of f, f(x)=y and g(y)=x. Not all functions have an inverse. HS.F-BF.B.3 HS.F-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with Mathematical Practices Essential Questions Create a graph and explain what transformation(s) were done on the parent function to create that graph. What are the transformations that can be done to a graph and how can they be represented algebraically? How do you determine if a graph is odd, even, or neither? Why are the two descriptions of an even function equivalent? Why are the two descriptions of an odd function equivalent? How do you determine if two functions are inverses of one another? Given a function, how do you find its inverse? Examples & Explanations HS.MP.4. Model with mathematics. Start by focusing on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept. While applying other transformations to a linear graph is appropriate at this level, it may be difficult for students to identify or distinguish between the effects of the other transformations included in this standard. Later in the year, extend this work to quadratic functions, and consider including absolute value functions. HS.MP.5. Use appropriate tools Students will apply transformations to functions and recognize functions as even and odd. Students 51 cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Connections: ETHS-S6C2-03; 11-12.WHST.2e strategically. HS.MP.7. Look for and make use of structure. may use graphing calculators or programs, spreadsheets, or computer algebra systems to graph functions. Examples: Compare the graphs of f(x)=3x with those of g(x)=3x+2 and h(x)=3x -1 to see that parallel lines have the same slope AND to explore the effect of the transformation of the function, f(x)=3x such that g(x)=f(x)+2 and h(x)=f(x) – 1. Explore the relationship between f(x)=3x, g(x)= 5x, and h( x) 1 x with a calculator to 2 develop a relationship between the coefficient on x and the slope. Describe the effect of varying the parameters a, h, and k on the shape and position of the (x + h) graph f(x) = ab + k, orally or in written format. What effect do values between 0 and 1 have? What effect do negative values have? Is f(x) = x - 3x + 2x + 1 even, odd, or neither? Explain your answer orally or in written format. Compare the shape and position of the graphs of f (x) = x 2 and g(x) = 2x 2 , and explain the differences in terms of the algebraic expressions for the functions. Describe the effect of varying the parameters a, h, and k have on the shape and position of 2 the graph of f(x) = a(x-h) + k. 3 2 52 Additional Domain Information – Building Functions (F-BF) Key Vocabulary Domain Range Rate of change Transformation Expand/contract Linear function Exponential function Quadratic function Example Resources Translate Recursive formula Explicit formula Even/odd function Inverse Books SpringBoard by College Board [activity 2.2, 4.2] Algebra I by McDougal and Littell [8.2, 8.5, 8.6] Developing Essential Understanding of Functions: Grades 9-12 by NCTM Technology http://www.khanacademy.org/#algebra-functions provides an extensive list of function video lectures http://www.purplemath.com/modules/index.htm provides resources about teaching functions http://a4a.learnport.org/page/comparing-functions provides multiple interactive applet investigations: introduction to recursive notation, etc. http://www.padowan.dk/graph/ A free downloadable graph generator www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons http://www.regentsprep.org/regents/math/algebra/AE7/PennyLabSheet.pdf Exponential growth and decay: Penny Activity http://illuminations.nctm.org/LessonDetail.aspx?ID=U142 Population of Trout Pond 4 lessons: recursion, numerical analysis, graphical analysis, and symbolic analysis. Common Student Misconceptions 2 Students have difficulty identifying that a variable squared with a coefficient does not mean to also square the coefficient. For example, 3x is 2 misrepresented as 9x . Students have difficulty applying rules of exponents when the exponent is rational. For example, adding/subtracting rational exponents that do not have like denominators. 2 Students confuse power functions and exponential functions. For example, students think that n is an exponential function because it contains an exponent. 53 Domain: Linear, Quadratic and Exponential Models (2 Clusters) Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 1 – Standards 1, 2 and 3) Construct and compare linear, quadratic, and exponential models and solve problems. Essential Concepts Linear functions grow by equal differences over equal intervals. Exponential functions grow by equal factors over equal intervals. Linear functions have an additive recursive pattern; exponential functions have a multiplicative recursive pattern. Linear and exponential functions can be constructed given a graph, a description of a relationship, or a set of input-output pairs (which may be given in a table). An exponential growth model will eventually exceed in quantity any linear or quadratic growth model. HS.F-LE.A.1 HS.F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions. Connections: ETHS-S6C2-03; SSHS-S5C5-03 a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Connection: 11-12.WHST.1a-1e b. Recognize situations in which one quantity changes at a constant Mathematical Practices HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in Essential Questions How do you determine if a given situation is modeled by a linear or exponential function? How do you construct an exponential function given a graph? Table? Description of a relationship? How do you know an exponential growth model will eventually exceed in quantity any linear or quadratic growth model? Examples & Explanations Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model and compare linear and exponential functions. Examples: A cell phone company has three plans. Graph the equation for each plan, and analyze the change as the number of minutes used increases. When is it beneficial to enroll in Plan 1? Plan 2? Plan 3? 1. $59.95/month for 700 minutes and $0.25 for each additional minute, 2. $39.95/month for 400 minutes and $0.15 for each additional minute, and 3. $89.95/month for 1,400 minutes and $0.05 for each additional minute. A computer store sells about 200 computers at the price of $1,000 per computer. For each $50 increase in price, about ten fewer computers are sold. How much should the computer store charge per computer in order to maximize their profit? Students can investigate functions and graphs modeling different situations involving simple and compound interest. Students can compare interest rates with different periods of compounding (monthly, daily) and compare them with the corresponding annual percentage rate. Spreadsheets and applets can be used to explore and model different interest rates and loan terms. A couple wants to buy a house in five years. They need to save a down payment of $8,000. They deposit $1,000 in a bank account earning 3.25% interest, compounded quarterly. How much will they need to save each month in order to meet their goal? 54 rate per unit interval relative to another. repeated reasoning. Connection: 11-12.RST.4 c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Sketch and analyze the graphs of the following two situations. What information can you conclude about the types of growth each type of interest has? o Lee borrows $9,000 from his mother to buy a car. His mom charges him 5% interest a year, but she does not compound the interest. o Lee borrows $9,000 from a bank to buy a car. The bank charges 5% interest compounded annually. Connections: ETHS-S6C1-03; ETHS-S6C2-03; 11-12.RST.4 HS.F-LE.A.2 HS.F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). Connections: ETHS-S6C1-03; ETHS-S6C2-03; 11-12.RST.4; SSHS-S5C5-03 Mathematical Practices HS.MP.4. Model with mathematics. HS.MP.8. Look for and express regularity in repeated reasoning. Examples & Explanations In constructing linear functions in F-LE.2, draw on and consolidate previous work in Grade 8 on th finding equations for lines and linear functions (8.EE.6, 8.F.4). In 8 grade students should do work with identifying slope and unit rates for linear functions given two points, a table or a graph. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to construct linear and exponential functions. Examples: x Determine an exponential function of the form f(x) = ab using data points from the table. Graph the function and identify the key characteristics of the graph. x 0 1 3 f(x) 1 3 27 Sara’s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in explicit form to describe the situation. x Solve the equation 2 = 300. x Possible solution using a graphing calculator: enter y = 2 and y = 300 into a graphing calculator and find where the graphs intersect, by viewing the table to see where the function values are about the same. 55 HS.F-LE.A.3 HS.F-LE.A.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Mathematic al Practices HS.MP.2. Reason abstractly and quantitatively. Examples & Explanations Start F-LE.3 by limiting to comparisons between linear and exponential models. Later in the year compare linear and exponential growth to quadratic growth. Example: x 2 Contrast the growth of the functions f(x)=3x, f(x)=3 and f(x) = x + 3. Linear, Quadratic and Exponential Models (F-LE) (Domain 3 – Cluster 2 – Standard 5) Interpret expressions for functions in terms of the situation they model. (Linear and exponential of the form f(x)=bx+k) Essential Concepts A given situation will set parameters for any linear or exponential function that models the situation. HS.F-LE.B.5 HS.F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of a context. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. Connections: ETHS-S6C1-03; ETHS-S6C2-03; SSHS-S5C5-03; 11-12.WHST.2e HS.MP.4. Model with mathematics. Essential Questions Create an example of a linear situation and give the function that can be used to model the situation. What does each part of the function represent in the context of the problem? What are the parameters for this function? Create an example of an exponential situation and give the function that can be used to model the situation. What does each part of the function represent in the context of the problem? What are the parameters for this function? Examples & Explanations x Limit exponential functions to those of the form f(x) = b + k. Students may use graphing calculators or programs, spreadsheets, or computer algebra systems to model and interpret parameters in linear, quadratic or exponential functions. Examples: The total cost for a plumber who charges $50 for a house call and $85 per hour would be expressed as the function y = 85x + 50. If the rate were raised to $90 per hour, how would the function change? x The equation y = 8,000(1.04) models the rising population of a city with 8,000 residents when the annual growth rate is 4%. o What would be the effect on the equation if the city’s population were 12,000 instead of 8,000? 56 o What would happen to the population over 25 years if the growth rate were 6% instead of 4%? n A function of the form f(n) = P(1 + r) is used to model the amount of money in a savings account that earns 5% interest, compounded annually, where n is the number of years since the initial deposit. What is the value of r? What is the meaning of the constant P in terms of the savings account? Explain either orally or in written format. Additional Domain Information – Linear, Quadratic and Exponential Models (F-LE) Key Vocabulary Linear function Exponential function Quadratic function Example Resources Books Explicit formula Even/odd function Principal Interest rate Growth/decay rate Algebra I by McDougal and Littell [9.8, 11.3] SpringBoard Algebra I by CollegeBoard [2.2] Developing Essential Understanding of Functions: Grades 9-12 by NCTM Technology http://www.ixl.com/math/algebra-1/write-linear-quadratic-and-exponential-functions Activities in creating functions based on data tables. http://a4a.learnport.org/page/comparing-functions Apply a linear and exponential model to a small data set in order to explore the differences in growth pattern, depending on which model is selected. http://www.padowan.dk/graph/ Free downloadable graph generator. eltrevoogmath.weebly.com/uploads/6/6/8/2/6682813/9-8.pdf This PDF is a note-taking guide for 9.8 McDougal Littell. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. http://www.illustrativemathematics.org/ This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons 57 http://www.nsa.gov/academia/_files/collected_learning/high_school/algebra/matchstick_math.pdf Use matchsticks and straws to create models of the three types of functions and analyze them. http://www.yummymath.com/2011/harry-potter/ Using the income and expense data from the Harry Potter films, create a function and predict the income of a hypothetical ninth movie. Technology Time SpringBoard [2.2 pg 80] Given 6 different types of functions, determine the domain and range of the functions. Investigation 3 Thinking with Mathematical Models, Connected Math 2 [pg 47] Inverse Variation Common Student Misconceptions 2 Students tend to draw all graphs of functions as linear. For example, they may graph x as 2x. Students misunderstand how to identify the parameters (function transformation) to build new functions from existing functions. For example, 2 2 2 to shift the graph of y = x left by 6 units, they may write y = (x - 6) rather than y = (x + 6) . Students often confuse the independent variable with the dependent variable. Students often have difficulty identifying the quantities to be represented by variables, in modeling problems,. Assessment Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments (e.g., pre-assessments, daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above. PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills, concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015. 58 High School Algebra 1 Conceptual Category: Statistics and Probability (1 Domain, 3 Clusters) Domain: Interpreting Categorical and Quantitative Data (3 Clusters) Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 1 – Standards 1, 2 and 3) Summarize, represent, and interpret data on a single count or measurement variable. Essential Concepts Sets of data can be represented on number lines via dot plots, histograms, and box plots, in order to look at and compare the overall shape of the data, measures of center and spread. Extreme data points (outliers) can affect the shape, measures of center, and spread of a given data set. The measure of center or variability that best interprets a data set will depend upon the shape of the data distribution and context of data collection. HS.S-ID.A.1 HS.S-ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots). Connections: SCHS-S1C1-04; SCHS-S1C2-03; SCHS-S1C2-05; SCHS-S1C4-02; SCHS-S2C1-04; ETHS-S6C2-03; SSHS-S1C1-04; 9-10.RST.7 Mathematical Practices HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. Essential Questions For a given data set, which measure of center or variability best describes the data and why? How can extreme data points affect the shape, measures of center and spread of a data set? What types of data would you want to display on a number line and why? Examples & Explanations In grades 6 – 8, students describe center and spread in a data distribution. Here they choose a summary statistic appropriate to the characteristics of the data distribution, such as the shape of the distribution or the existence of extreme data points. A statistical process is a problem-solving process consisting of four steps: 1. formulating a question that can be answered by data; 2. designing and implementing a plan that collects appropriate data; 3. analyzing the data by graphical and/or numerical methods; 4. and interpreting the analysis in the context of the original question. Example: What measure of center or variability would best represent the data distribution for the height of basketball players on this team? Why? Are there any extreme data points that may skew the data? Basketball Team – Height of Players in inches for 2010-2011 Season 75, 73, 76, 78, 79, 78, 79, 81, 80, 82, 81, 84, 82, 84, 80, 84 59 HS.S-ID.A.2 HS.S-ID.A.2 Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. Connections: SCHS-S1C3-06; ETHS-S6C2-03; SSHS-S1C1-01 Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics.HS. MP.5. Use appropriate tools strategically. HS.S-ID.A.3 HS.S-ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Examples & Explanations Students may use spreadsheets, graphing calculators and statistical software for calculations, summaries, and comparisons of data sets. Examples: The two data sets below depict the housing prices sold in the King River area and Toby Ranch areas of Pinal County, Arizona. Based on the prices below, which price range can be expected for a home purchased in Toby Ranch? In the King River area? In Pinal County? o King River area {1.2 million, 242000, 265500, 140000, 281000, 265000, 211000} o Toby Ranch homes {5million, 154000, 250000, 250000, 200000, 160000, 190000} Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and standard deviation. Explain how the values vary about the mean and median. What information does this give the teacher? HS.MP.7. Look for and make use of structure. Mathematical Practices HS.MP.2. Reason abstractly and quantitatively. Examples & Explanations Students may use spreadsheets, graphing calculators and statistical software to statistically identify outliers and analyze data sets with and without outliers as appropriate. Comparing two data sets using a histogram. Not only can the shape of the distribution be observed, but so can the distribution's location and spread. Figure 16 shows how a mean has increased -- a transition from the distribution shown at the left (blue) to the one shown on the right (green). Figure 60 HS.MP.3. Construct viable arguments and critique the reasoning of others. 17 shows a different method of comparing distributions. The original data set (shown in green) has greater variability than the later data set (the blue histogram superimposed over the original data set). HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. Figure 16: Histogram of two data sets, one with increased mean Figure 17: Histogram of two data sets, one with increased variabilityFrom: http://illuminae.info/matec/index.php?option=com_content&view=article&id=12:quality-tools-andspc-charts&catid=9 61 Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 2 – Standards 5 and 6) Summarize, represent, and interpret data on two categorical and quantitative variables. (Linear focus, discuss general principles) Essential Concepts Two-way frequency tables can be used to interpret joint, marginal and conditional relative frequencies of categorical data. Two-way frequency tables and scatter plots of categorical data can be used to identify possible associations and trends in the data. Scatter plots of data sets can be used to identify the type of function that best represents the shape of the data (linear, quadratic or exponential). Residuals (lines of regressions) are drawn on scatter plots in order to informally assess the fit of a function to a data set. If a scatter plot has a linear association, then a line of best fit can be drawn to interpret the data set. HS.S-ID.B.5 HS.S-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. Connections: ETHS-S1C2-01; ETHS-S6C2-03; 11-12.RST.9; 11-12.WHST.1a-1b; 11-12.WHST.1e Mathematical Practices HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. How are two-way frequency tables used to interpret joint, marginal and conditional relative frequencies of categorical data? How do you use a two-way frequency table or scatter plot to identify associations or trends in a data set? Why would you want to identify trends or associations in a data set? Why would you want to informally assess and identify a type of function to fit a data set? Examples & Explanations Students may use spreadsheets, graphing calculators, and statistical software to create frequency tables and determine associations or trends in the data. Examples: HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. Essential Questions Two-way Frequency Table A two-way frequency table is shown below displaying the relationship between age and baldness. We took a sample of 100 male subjects, and determined who is or is not bald. We also recorded the age of the male subjects by categories. Bald No Yes Total Two-way Frequency Table Age Younger than 45 45 or older 35 11 24 30 59 41 Total 46 54 100 The total row and total column entries in the table above report the marginal frequencies, while entries in the body of the table are the joint frequencies. Two-way Relative Frequency Table The relative frequencies in the body of the table are called conditional relative frequencies. 62 HS.MP.5. Use appropriate tools strategically. HS.S-ID.B.6 HS.S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Connections: SCHS-S1C2-05; SCHS-S1C3-01; ETHS-S1C2-01; ETHS-S1C3-01; ETHS-S6C2-03 a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. Connection: 11-12.RST.7 b. Informally assess the fit of a function by plotting and analyzing residuals. Bald HS.MP.8. Look for and express regularity in repeated reasoning. Mathematic al Practices HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. No Yes Total Two-way Relative Frequency Table Age Total Younger than 45 45 or older 0.35 0.11 0.46 0.24 0.30 0.54 0.59 0.41 1.00 Examples & Explanations Students take a more sophisticated look at using a linear function to model the relationship between two numerical variables. In addition to fitting a line to data, students assess how well the model fits by analyzing residuals. S.ID.6b should be focused on linear models, but may be used to preview quadratic functions for later in the year. This concept can be explained through the use of technology to model the idea and allow students to explore this standard. The residual in a regression model is the difference between the observed and the predicted y for some x (where y is the dependent variable and x is the independent variable). So if we have a model y = ax + b and a data point (x i, y i ) , the residual for this point is ri = y i - (ax i + b) . Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals. Example: Measure the wrist and neck size of each person in your class and make a scatter plot. Find the least squares regression line. Calculate and interpret the correlation coefficient for this linear regression model. Graph the residuals and evaluate the fit of the linear equation. Use the line of best fit to predict the wrist size for a person not in your class. 63 Connections: 11-12.RST.7; 11-12.WHST.1b-1c c. Fit a linear function for a scatter plot that suggests a linear association. HS.MP.8. Look for and express regularity in repeated reasoning. Connection: 11-12.RST.7 Interpreting Categorical and Quantitative Data (S-ID) (Domain 1 – Cluster 3 – Standards 7, 8 and 9) Interpret linear models Essential Concepts If a scatter plot has a linear association, then a linear model can be drawn and used to identify and interpret the meaning of the slope (constant rate of change) and the intercept (constant term) between the data sets. Technology is used to compute and interpret the correlation coefficient (the slope) of a linear model. A correlation does not necessarily mean there is causation. HS.S-ID.C.7 HS.S-ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Connections: SCHS-S5C2-01; ETHS-S1C2-01; ETHS-S6C2-03; 9-10.RST.4; 9-10.RST.7; 9-10.WHST.2f Mathematical Practices HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. Essential Questions How do you interpret the meaning of a slope in a linear model in context? What is the meaning of an intercept in terms of a linear model for a given data set? How are the slope and correlation coefficient related? How do you use technology to compute the correlation coefficient? What is the difference between a correlation and causation? Give an example of a relationship that has a correlation but is not causation and explain why. Examples & Explanations Build on students’ work with linear relationships in eighth grade and introduce the correlation coefficient. Students may use spreadsheets or graphing calculators to create representations of data sets and create linear models. Example: Lisa lights a candle and records its height in inches every hour. The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the candle’s height (h) as a function of time (t) and state the meaning of the slope and the intercept in terms of the burning candle. Solution: h = -1.7t + 20 64 HS.MP.5. Use appropriate tools strategically. HS.S-ID.C.8 HS.S-ID.C.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. Connections: ETHS-S1C2-01; ETHS-S6C2-03; 11-12.RST.5; 11-12.WHST.2e HS.S-ID.C.9 HS.S-ID.C.9 Distinguish between correlation and causation. Connection: 9-10.RST.9 Slope: The candle’s height decreases by 1.7 inches for each hour it is burning. Intercept: Before the candle begins to burn, its height is 20 inches. HS.MP.6. Attend to precision. Mathematical Practices HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.8. Look for and express regularity in repeated reasoning. Mathematical Practices HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. HS.MP.6. Attend to precision. Examples & Explanations The focus here is on the computation and interpretation of the correlation coefficient as a measure of how well the data fit the relationship. Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals and correlation coefficients. Example: Collect height, shoe-size, and wrist circumference data for each student. Determine the best way to display the data. Answer the following questions: Is there a correlation between any two of the three indicators? Is there a correlation between all three indicators? What patterns and trends are apparent in the data? What inferences can be made from the data? Examples & Explanations The important distinction between a statistical relationship and a cause-and-effect relationship arises in S-ID.9. Some data leads observers to believe that there is a cause and effect relationship when a strong relationship is observed. Students should be careful not to assume that correlation implies causation. The determination that one thing causes another requires a controlled randomized experiment. Example: Diane did a study for a health class about the effects of a student’s end-of-year math test scores on height. Based on a graph of her data, she found that there was a direct relationship between students’ math scores and height. She concluded “doing well on your end-of-year math tests makes you tall.” Is this conclusion justified? Explain any flaws in Diane’s reasoning. 65 Additional Domain Information – Interpreting Categorical and Quantitative Data (S-ID) Key Vocabulary Outliers Quartile intervals Frequency Two-way frequency tables Correlation coefficient Standard deviation Interval Scatter plot Causation Line of regression (residual) Data distribution Line of best fit Distribution Correlation Example Resources Books Textbooks (Pending) Focus in High School Mathematics: Reasoning and Sense Making in Statistics and Probability, National Council of Teachers of Mathematics Publication Technology http://www.shodor.org/interactivate/lessons/LinearRegressionCorrelation/ Within a complete lesson with categorical and quantitative data concepts, with vocabulary, there is a useful tool for scatter plots where the student is able to plot the data, find the best line of fit, and analyze residuals. http://nlvm.usu.edu/en/nav/topic_t_5.html This page contains a variety of data analysis formats to present, analyze and predict data. www.classzone.com/ This is the site to access the book and extra resources online. http://illuminations.nctm.org/ This site has a bank of different lessons published by NCTM. http://www.hippocampus.org/ This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. http://www.wolframalpha.com/ This site is a resource for looking up mathematical facts and also provides a place to solve and graph www.Illustrativemathematics.org – This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons www.malamaaina.org/files/mathematics/lesson7.pdf This lesson is a series of activities designed to use previous data gathered at a higher level with each succeeding activity. http://www.indiana.edu/~iucme/mathmodeling/lessons.htm - A series of 40 problem-based activities to develop data-gathering techniques th th from 7 – 12 grade. Common Student Misconceptions Students confuse the measures of center and how an outlier can have different impacts on the mean and median. For example, students think that including an outlier changes the median, when in fact it changes the mean. Students tend to have difficulty with the distinction between experimental and predicted values. Students sometimes have difficulty connecting lines to functions. For example, they may have difficulty in predicting that a steeper line will change more quickly than a gradual one. 66 Students have more difficulty with negative correlations than with positive correlations. Students may have difficulty identifying when a group of data represent a linear or nonlinear function. Students have difficulty placing lines of best fit. For example, they may try to connect all of the points on a scatter plot, or may not draw the line through the middle of the data. Students may believe that a correlation implies causation. Assessment Both formative and summative assessments are vital components of effective mathematics curricula. Formative assessments, (e.g., pre-assessments, daily checks for understanding, discussions of strategies students use to solve problems, etc.) assist in instructional planning and implementation; summative assessments (e.g., unit assessments, quarterly benchmarks, etc.) inform learner growth related to important mathematics concepts. All district-adopted resources contain multiple assessment tools and include online resources that can be used for the purposes delineated above. PARCC also will provide two end of the year summative assessments. The first, a performance-based assessment, will focus on applying skills, concepts, and understandings to solve multi-step problems requiring abstract reasoning, precision, perseverance, and strategic use of tools. The performance measure will be administered as close to the end of the school year as possible. The second, an end of the year machine-scorable summative assessment, will be administered after approximately 90% of the school year. These assessments are to begin during the SY 2014-2015. 67 68 69 70 71 72 73 74 75 76 77 78