STANDARDS FOR MATHEMATICS. High School Algebra 1

Transcription

1 STANDARDS FOR MATHEMATICS High School Algebra 1 1

2 Number and Quantity The Real Number System (N-RN) Quantities (N-Q) The Complex Number System (N-CN) Vector and Matrix Quantities (N-VM) High School Overview Conceptual Categories and Domains 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) Statistics and Probability Interpreting Categorical and Quantitative Data (S- ID) Making Inferences and Justifying Conclusions (S- IC) Conditional Probability and the Rules of Probability (S-CP) Using Probability to Make Decisions (S-MD) Contemporary Mathematics Discrete Mathematics (CM-DM) Functions Interpreting Functions (F-IF) Building Functions (F-BF) Linear, Quadratic, and Exponential Models (F-LE) Trigonometric Functions (F-TF) Geometry 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) (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. Modeling 2

3 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. 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.! (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. 3

4 Seeing Structure in Expressions (A-SSE) Interpret the structure of expressions Write expressions in equivalent forms to solve problems 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 High School - Algebra Overview! (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. 4

5 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 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! (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. 5

6 Congruence (G-CO) Experiment with transformations in the plane Understand congruence in terms of rigid motions Prove geometric theorems Make geometric constructions 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 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 High School Geometry Overview! 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 (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. 6

7 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 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 Statistics and Probability Overview! (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. High School Contemporary Mathematics Overview! Discrete Mathematics (CM-DM) Understand and apply vertex-edge graph topics 7

8 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

9 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 CO 2 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 Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol ( ). 9

10 Standards for Practice: High School Standards for Practice Standards Explanations and Examples Students are expected to: 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. HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. 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

11 Standards for Practice Standards Explanations and Examples Students are expected to: 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. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.4. Model with mathematics. 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

12 Standards for Practice Standards Explanations and Examples Students are expected to: 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. HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. HS.MP.7. Look for and make use of structure. 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. By high school, students look closely to discern a pattern or structure. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as 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 composed of several objects. For example, they can see 5 3(x y) 2 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

13 Standards for Practice Standards Explanations and Examples Students are expected to: 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. HS.MP.8. Look for and express regularity in repeated reasoning. 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 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + 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

14 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 1 - Standards 1 and 2) Extend the properties of exponents to rational exponents. Essential Concepts Rational exponents are exponents that are fractions. Properties of integer exponents extend to properties of rational exponents. Properties of rational exponents are used to simplify and create equivalent forms of numerical expressions. Rational exponents can be written as radicals, and radicals can be written as rational exponents. HS.N-RN.1 HS.N-RN.1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. Essential Questions How do you use properties of rational exponents to simplify and create equivalent forms of numerical expressions? Why are rational exponents and radicals related to each other? Given an expression with a rational exponent, how do you write the equivalent radical expression? In implementing the standards in curriculum, these standards should occur before discussing exponential functions with continuous domains. Students may explain orally or in written format. Example: For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. HS.N-RN.2 HS.N-RN.2. Rewrite expressions involving radicals and rational exponents using the properties of exponents. HS.MP.7. Look for and make use of structure. Examples: = 5 ; = (Continued on next page)

15 Rewrite using fractional exponents: x Rewrite in at least three alternate forms. 2 x Solution: 2 x = = = x x x x Rewrite Using only rational exponents. 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.3 HS.N-RN.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 HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others = 2 = 2 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. 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 Rational exponents 15

16 Example Resources Books Building Powerful Numeracy for Middle and High School Students, by Pamela Weber Harris. This contains supplementary resources for the Arizona adopted math books. Technology - Provides teachers or students with virtual manipulatives to interact with the concepts. - Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. Help with graphing calculators and rational functions. This site has a bank of different lessons published by NCTM This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons For Fractional Exponents: This is a short video lesson on converting radicals to fractional exponent notation. 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. This lesson includes laws of exponents but moves on to basic equations with rational exponents of the form x 3 5 = 1 8 For Operations Bringing Together Rational and Irrational Numbers: 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 27 3 = = 9 instead of 27 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. 16

17 m n Students may have difficulty converting between radical notation and fractional exponent notation: b = and the n. n 1 b m. They might confuse the m Students tend to assume that they can combine integers and radical expressions: Example, 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. 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.1 HS.N-Q.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 HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools = 6. They don t 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? 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. (Continued on next page)

18 origin in graphs and data displays. HS.N-Q.2 HS.N-Q.2 Define appropriate quantities for the purpose of descriptive modeling. strategically. HS.MP.6. Attend to precision. HS.MP.4. Model with mathematics. 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 sec 1 mile 5280 ft 60sec 1 min 60min = 1 hr mile 5280 hr 8.18 miles per hour. 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: 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.N-Q.3 HS.N-Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. HS.MP.6. Attend to precision. Mathematic al HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. 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 Unit rate Descriptive model Ratio Scale Equivalent Origin Unit Conversion 18

19 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 Useful for checking correct conversions. Useful site with virtual practice problems. This is the site to access the book and extra resources online. This site has a bank of different lessons published by NCTM This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. This is a webpage that has the new standards with sample classroom tasks linked to some of the standards. Example Lessons A nicely thorough introduction to the basics of unit conversion, with practice problems at the end. The video lesson describes how to use unit conversion to solve word problems, labeling each step of the process carefully. 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 5280 ft Students often have difficulty understanding how ratios expressed in different units can be equal to one. For example, is simply one, 1 mile and it is permissible to multiply by that ratio. 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. 19

20 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

21 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.1 HS.A-SSE.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, HS.MP.1. Make sense of problems and persevere in solving them. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with 21 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? 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: Interpret P(1+r) n as the product of P and a factor not depending on P. (Continued on next page)

22 interpret P(1+r) n as the product of P and a factor not depending on P. mathematics. HS.MP.7. Look for and make use of structure. Suppose the cost of cell phone service for a month is represented by the expression 0.40s Students can analyze how the coefficient of 0.40 represents the cost of one minute (40 ), while the constant of 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. Factor 3x(x 5) + 2(x 5). HS.A-SSE.2 HS.A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Mathematic al HS.MP.2. Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure. 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). 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. See x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). 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 Essential Questions The solutions of quadratic equations are the x-intercepts of the parabola What are the solutions to a quadratic equation and how do they relate or zeros of quadratic functions. to the graph? Factoring methods and the method of completing the square reveal What attributes of the graph will factoring and completing the square attributes of the graphs of quadratic functions. reveal about a quadratic function? Factoring a quadratic reveals the zeros of the function. How are properties of exponents used to transform expressions for Completing the square in a quadratic equation reveals the maximum or exponential functions? minimum value of the function. Why would you want to transform an expression for an exponential Properties of exponents are used to transform expressions for function? exponential functions. 22

23 HS.A-SSE.3 HS.A-SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Connections: 9-10.WHST.1c; 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. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 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 for and make use of structure. 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. 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). A pair of coordinates (h, k) from the general form f(x) = a(x h) 2 + k represents the vertex of the parabola, where h represents a horizontal shift and k represents a vertical shift of the parabola y = x 2 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. Examples: Express 2(x 3 3x 2 + x 6) (x 3)(x + 4) in factored form and use your answer to say for what values of x the expression is zero. The expression 1.15 t can be rewritten as (1.15 1/12 ) 12t t to reveal the approximate 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 (2 x ) (3 x ) 2 3 ( x ) 23

24 Additional Domain Information Seeing Structure in Expressions (A-SSE) Key Vocabulary Expression Term Coefficient Factor Exponent Base Simplify Greatest Common Factor Quadratic Polynomial Binomial 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 online software to create visuals Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. This is the site to access the book and extra resources online. This site has a bank of different lessons published by NCTM. This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons 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. 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. 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. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Students will identify linear and nonlinear relationships in a variety of contexts. 24

25 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. Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x 2 rather than 5x. Students will forget to distribute to all terms when multiplying. For example, 6 (2x + 1) = 12x + 1 rather than 12x + 6. Students may not follow the Order of Operations when simplifying expressions. For example, 4x 2 when x = 3 may be incorrectly evaluated as = 12 2 = 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. Students fail to use the property of exponents correctly when using the distributive property. For example, 3x(2x 1) = 6x 3x = 3x instead of simplifying as 3x ( 2x 1) = 6x 2 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 = 12. In addition, students commonly misevaluate 3 2 = 9 rather than 3 2 = 9. Students routinely see 3 2 as the same as ( 3) 2 = 9. A method that may clear up the misconception is to have students rewrite as x 2 = 1 x 2 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 example, students will often simplify (x 2 ) 3 = x 5 instead of x 6. 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. 25

26 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. HS.A-APR.1 HS.A-APR.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. Connection: 9-10.RST.4 HS.MP.8. Look for regularity in repeated reasoning. 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. Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x. 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 Term Coefficient Simplify Exponent Base Polynomial Binomial Factor Distributive Property Trinomial Linear Exponential Quadratic Closure Property 26

27 Example Resources Books Textbook Focus in High School Mathematics: Reasoning and Sense Making (Algebra), Chapter 3: Formal Algebra The Xs and Whys of Algebra: Key Ideas and Common Misconceptions Technology Key Curriculum Press, Exploring Algebra I with the Geometer s Sketchpad online software to create visuals Khan Academy contains useful video lessons and also a galaxy of practice modules that enable students to check answers online. This is the site to access the book and extra resources online. This site has a bank of different lessons published by NCTM. This site has online videos and interactive lessons for both teachers and students to use and can be matched to a particular textbook. This site is a resource for looking up mathematical facts and also provides a place to solve and graph equations. This is a webpage that has the new standards with sample classroom tasks linked to some of the standards Example Lessons 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. 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. 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. Students will change the degree of the variable when adding/subtracting like terms. For example, 2x + 3x = 5x 2 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) = x 2 6 rather than x 2 2x + 3x 6. 27