Algebra 1 (#1200310) 2017 2018 Course Description Course Title: Algebra 1 Course Number: 1200310 Version Description: The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, called units, deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Standards for Mathematical Practice apply throughout each course, and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. Unit 1- Relationships Between Quantities and Reasoning with Equations: By the end of eighth grade students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. All of this work is grounded on understanding quantities and on relationships between them. SKILLS TO MAINTAIN: Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds on this understanding. Unit 2- Linear and Exponential Relationships: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Unit 3- Descriptive Statistics: This unit builds upon students prior experiences with data, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe and approximate linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit. Unit 4- Expressions and Equations: In this unit, students build on their knowledge from unit 2, where they extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving quadratic expressions.
General Notes: Unit 5- Quadratic and Modeling: In this unit, students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions, absolute value, step, and those that are piece wise-defined. Fluency Recommendations A/G- Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). A-APR.1- Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in Algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. A-SSE.1b- Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square, and other mindful algebraic calculations. English Language Development ELD Standards Special Notes Section: Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL's need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link: http://www.cpalms.org/uploads/docs/standards/eld/ma.pdf For additional information on the development and implementation of the ELD standards, please contact the Bureau of Student Achievement through Language Acquisition at sala@fldoe.org. Modeling standards are marked with a star/asterisk at the end of the standard. This denotes that it is a modeling standard from the Modeling conceptual category. Modeling is best interpreted not as a collection of isolated topics but 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. It is important to note that there are 61 specific modeling standards throughout the high school standards. Look for a star/asterisk in the course descriptions to delineate.
66 Florida Standards: 10 Language Arts, 2 ELD, 46 Mathematics, 8 Standards for Mathematical Practice 10 Language Arts Florida Standards / 2 English Language Development Standards Textbook Section Initiate and participate effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grades 9 10 topics, texts, and issues, building on others ideas and expressing their own clearly and persuasively. a. Come to discussions prepared, having read and researched material under study; explicitly draw on that preparation by referring to evidence from texts and other research on the topic or issue to stimulate a thoughtful, well-reasoned exchange of ideas. b. Work with peers to set rules for collegial discussions and decision-making (e.g., LAFS.910.SL.1.1: informal consensus, taking votes on key issues, presentation of alternate views), clear goals and deadlines, and individual roles as needed. c. Propel conversations by posing and responding to questions that relate the current discussion to broader themes or larger ideas; actively incorporate others into the discussion; and clarify, verify, or challenge ideas and conclusions. d. Respond thoughtfully to diverse perspectives, summarize points of agreement and disagreement, and, when warranted, qualify or justify their own views and understanding and make new connections in light of the evidence and reasoning presented. Integrate multiple sources of information presented in diverse media or formats (e.g., LAFS.910.SL.1.2: visually, quantitatively, orally) evaluating the credibility and accuracy of each source. Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, LAFS.910.SL.1.3: identifying any fallacious reasoning or exaggerated or distorted evidence. Present information, findings, and supporting evidence clearly, concisely, and logically LAFS.910.SL.2.4: such that listeners can follow the line of reasoning and the organization, development, substance, and style are appropriate to purpose, audience, and task. Follow precisely a complex multistep procedure when carrying out experiments, taking LAFS.910.RST.1.3: measurements, or performing technical tasks, attending to special cases or exceptions defined in the text. Determine the meaning of symbols, key terms, and other domain-specific words and LAFS.910.RST.2.4: phrases as they are used in a specific scientific or technical context relevant to grades 9 10 texts and topics. Translate quantitative or technical information expressed in words in a text into visual LAFS.910.RST.3.7: form (e.g., a table or chart) and translate information expressed visually or mathematically (e.g., in an equation) into words.
Write arguments focused on discipline-specific content. a. Introduce precise claim(s), distinguish the claim(s) from alternate or opposing claims, and create an organization that establishes clear relationships among the claim(s), counterclaims, reasons, and evidence. b. Develop claim(s) and counterclaims fairly, supplying data and evidence for each while pointing out the strengths and limitations of both claim(s) and counterclaims in a discipline-appropriate form and in a manner that anticipates LAFS.910.WHST.1.1: the audience s knowledge level and concerns. c. Use words, phrases, and clauses to link the major sections of the text, create cohesion, and clarify the relationships between claim(s) and reasons, between reasons and evidence, and between claim(s) and counterclaims. d. Establish and maintain a formal style and objective tone while attending to the norms and conventions of the discipline in which they are writing. e. Provide a concluding statement or section that follows from or supports the argument presented. Produce clear and coherent writing in which the development, organization, and style LAFS.910.WHST.2.4: are appropriate to task, purpose, and audience. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research. English language learners communicate information, ideas and concepts necessary for ELD.K12.ELL.MA.1: academic success in the content area of Mathematics. English language learners communicate for social and instructional purposes within the ELD.K12.ELL.SI.1: school setting. 46 Mathematics Florida Standards Section Understand that polynomials form a system analogous to the integers, namely, they are Math Nation 1.1, 3.3, 3.4, 3.5 closed under the operations of addition, subtraction, and multiplication; add, subtract, MAFS.912.A-APR.1.1: and multiply polynomials. Larson 8.1, 8.2, 8.3, 8.3 pt. 2 Algebra Arithmetic Remarks/Examples: with Polynomials and Algebra 1 - Fluency Recommendations Rational Expressions Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. MAFS.912.A-APR.2.3: Algebra Arithmetic with Polynomials and Rational Expressions Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Math Nation 8.10, 8.12 Larson 8.4, 8.6, 8.7, 8.8
MAFS.912.A-CED.1.1: Algebra Create Equations that Describe Numbers or Relationships MAFS.912.A-CED.1.2: Algebra Create Equations that Describe Numbers or Relationships MAFS.912.A-CED.1.3: Algebra Create Equations that Describe Numbers or Relationships MAFS.912.A-CED.1.4: Algebra Create Equations that Describe Numbers or Relationships MAFS.912.A-REI.1.1: Algebra Reasoning with Equations and MAFS.912.A-REI.2.3: Algebra Reasoning with Equations and Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational, absolute, and exponential functions. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 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. 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. 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. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Math Nation 2.3, 2.5, 2.6, 8.5 Larson 1.1, 1.2, 1.3, 1.4, 1.5, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 8.4, 8.5, 8.6, 8.7, 8.8, 9.4, 9.5, 9.6 pt. 2 Math Nation 2.8, 2.9, 4.3, 4.4, 6.1, 8.5 Larson 1.7, 3.2, 3.3, 3.5, 3.6, 3.7, 4.1, 4.2, 4.3, 4.4, 4.6, 4.7, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 7.4, 7.5, 9.1, 9.2, 9.3, 9.4, 9.8 Math Nation 4.2, 4.3, 4.4, 4.6, 4.8, 4.9, 4.10, 8.5 Larson 1.7, 3.2, 3.3, 3.5, 3.6, 4.1, 4.4, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7, 6.1, 6.2, 6.3, 6.4, 6.5, 6.6, 9.1, 9.2, 9.3 Math Nation 2.8 Larson 2.8 Math Nation 2.2, 2.3 Larson 2.1 Ext., 2.2, 2.5 Ext. Math Nation 2.1, 2.2, 2.3, 2.5, 2.6, 2.7, 2.8 Larson 1.1, 1.2, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 3.5 Ext, 5.1, 5.2, 5.3, 5.4
MAFS.912.A-REI.2.4: Algebra Reasoning with Equations and MAFS.912.A-REI.3.5: Algebra Reasoning with Equations and MAFS.912.A-REI.3.6: Algebra Reasoning with Equations and MAFS.912.A-REI.4.10: Algebra Reasoning with Equations and MAFS.912.A-REI.4.11: Algebra Reasoning with Equations and MAFS.912.A-REI.4.12: Algebra Reasoning with Equations and MAFS.912.A-SSE.1.1: Algebra Seeing Structure in Expressions MAFS.912.A-SSE.1.2: 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 p)² = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x² = 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. 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. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 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). Explain why the x-coordinates of the points where the 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. 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. Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1 + r) n as the product of P and a factor not depending on P. Use the structure of an expression to identify ways to rewrite it. For example, see x 4 - y 4 as (x²)² (y²)², thus recognizing it as a difference of squares that can be factored as (x² y²)(x² + y²). Math Nation 2.4, 5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10, 6.2, 8.5 Larson 8.4, 8.5, 8.6, 8.7, 8.8, 9.4, 9.5, 9.5 Ext. 9.6, 9.6 pt. 2 Math Nation 4.7 Larson 6.2, 6.4, 6.5 Math Nation 4.5, 4.6, 4.7, 4.8 Larson 6.1, 6.2, 6.3, 6.4, 6.5 Math Nation 2.9, 4.2, 4.3, 4.4, 4.5, 4.6 Larson 3.2, 5.3 Ext. Math Nation 4.6, 6.9, 8.14 Larson 2.5, 3.5 Ext., 3.7, 9.3, 9.4, 9.7 Math Nation 4.9, 4.10 Larson 5.7, 6.6 Math Nation 1.1, 1.2, 3.3, 3.4, 6.5 Larson 1.2, 1.3 Math Nation 1.3, 1.4, 1.5, 2.2, 3.3, 3.4, 5.2, 5.5, 5.6
Algebra Seeing Structure in Expressions MAFS.912.A-SSE.2.3: Algebra Seeing Structure in Expressions MAFS.912.F-BF.1.1: Building MAFS.912.F-BF.2.3: Building MAFS.912.F-IF.1.1: Interpreting MAFS.912.F-IF.1.2: Interpreting Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 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 t/12 ) 12t 1.012 12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. 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 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. 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). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Larson 8.3, 8.7 Math Nation 5.2, 5.3, 5.4, 5.5, 5.10, 7.3, 7.4 Larson 2.1, 7.1, 7.2, 7.3, 8.5, 8.6, 8.8, 9.5 Math Nation 3.6, 4.1, 7.1, 8.4 Larson 4.1, 8.2, 8.2 pt. 2 Math Nation 3.10, 6.7, 6.8, 7.6, 8.13 Larson 3.7, 4.1, 5.5 Ext., 7.4, 7.5, 9.1, 9.2 Ext., 9.5 Ext. Math Nation 3.1, 3.2, 8.6 Larson 1.7, 1.8, 3.7 Math Nation 3.1, 3.2, 8.6 Larson 3.7
MAFS.912.F-IF.1.3: Interpreting MAFS.912.F-IF.2.4: Interpreting MAFS.912.F-IF.2.5: Interpreting MAFS.912.F-IF.2.6: Interpreting MAFS.912.F-IF.3.7: Interpreting MAFS.912.F-IF.3.8: Interpreting 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. 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 maximums and minimums; symmetries; end behavior; and periodicity. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 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. 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 & quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude, and using phase shift. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. 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. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (0.97) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. Math Nation 4.1, 7.1, 8.3, 8.4 Larson 4.3 Ext, 7.5 Ext. A, 7.5 Ext. B Math Nation 3.7, 3.8, 5.1, 6.1, 7.3, 7.4, 8.1, 8.2 Larson 1.8, 3.3, 3.4, 3.5, 4.1, 4.2, 4.3, 4.4, 4.6, 4.7 Ext., 9.1, 9.3, 9.8, 9.9 Math Nation 3.2, 8.5 Larson 1.8, 3.1, 3.2, 3.2 Ext., 3.3, 3.5, 3.7, 4.4, 9.1 Math Nation 3.9, 8.1, 8.2 Larson 3.4, 3.6, 4.2, 4.3, 9.9 Math Nation 6.3, 6.4, 6.5, 6.6, 7.3, 7.4, 8.5, 8.6, 8.7, 8.8, 8.9, 8.11, 8.12 Larson 1.8, 3.2, 3.3, 3.5, 3.6, 3.7, 4.3, 5.5 Ext., 6.5 Ext., 7.4, 7.5, 8.1, 9.2, 9.2 Ext., 9.3, 9.5 Ext., 9.1, 9.2, 9.8 Math Nation 5.10, 6.1, 6.4, 6.5, 6.6, 7.3, 7.4, 7.5 Larson 8.4, 8.5,8.6, 9.5 Ext.
MAFS.912.F-IF.3.9: Interpreting MAFS.912.F-LE.1.1: Linear, Quadratic, and Exponential Models MAFS.912.F-LE.1.2: Linear, Quadratic, and Exponential Models MAFS.912.F-LE.1.3: Linear, Quadratic, and Exponential Models MAFS.912.F-LE.2.5: Linear, Quadratic, and Exponential Models MAFS.912.N-Q.1.1: Number and Quantity Quantities MAFS.912.N-Q.1.2: Number and Quantity Quantities 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. Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Interpret the parameters in a linear or exponential function in terms of a context. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Define appropriate quantities for the purpose of descriptive modeling. Remarks/Examples: Algebra 1 Content Notes: Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions. Math Nation 6.6, 8.1, 8.2 Larson 9.9 Math Nation 8.1, 8.2, 8.3 Larson 7.4, 7.5, 7.5 pt. 2, 9.8, 9.9 Math Nation 4.1, 4.3, 7.1, 7.2 Larson 1.8, 4.1, 4.2, 4.3, 4.4, 4.5, 7.4, 7.5, 7.5 Ext. Math Nation 8.1, 8.2, 8.3 Larson 7.5 pt. 2, 9.8, 9.9 Math Nation 4.2, 4.3, 4.6, 4.9, 7.5 Larson 4.1, 7.4, 7.5, 9.8 Math Nation Incorporated Throughout Larson 1.1, 1.3, 1.6, 1.8, 2.1, 2.8, 3.2 Math Nation Incorporated Throughout Larson 1.6, 2.1 MAFS.912.N-Q.1.3: Choose a level of accuracy appropriate to limitations on measurement when reporting Math Nation Incorporated Number and Quantity Throughout quantities. Quantities
MAFS.912.N-RN.1.1: Number and Quantity The Real Number System MAFS.912.N-RN.1.2: Number and Quantity The Real Number System MAFS.912.N-RN.2.3: Number and Quantity The Real Number System MAFS.912.S-ID.1.1: MAFS.912.S-ID.1.2: MAFS.912.S-ID.1.3: 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. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Represent data with plots on the real number line (dot plots, histograms, and box plots). Remarks/Examples: 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. 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. Remarks/Examples: 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. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Remarks/Examples: 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. Larson 1.6 Math Nation 1.6 Larson 7.3, 7.3 Ext Math Nation 1.6, 1.7, 1.8 Larson 7.3 Math Nation 1.9 Larson 2.1 Math Nation 9.1, 9.2, 9.3, 9.4 Larson 10.4, 10.5 Math Nation 9.5, 9.6, 9.7, 9.8 Larson 10.2, 10.2 Ext. 10.4, 10.5 Math Nation 9.9 Larson 10.2, 10.2 Ext., 10.4, 10.5, 10.5 Ext.
MAFS.912.S-ID.2.5: MAFS.912.S-ID.2.6: MAFS.912.S-ID.3.7: MAFS.912.S-ID.3.8: 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. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 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, and exponential models. b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Remarks/Examples: 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. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Compute (using technology) and interpret the correlation coefficient of a linear fit. Math Nation 10.1, 10.2, 10.3 Larson 10.3 Math Nation 10.4, 10.5, 10.6, 10.7 Larson 1.8, 4.6, 4.7, 4.7 Ext., 9.8 Math Nation 4.2, 4.3, 4.4, 10.4 Larson 3.4, AL2-3, 4.1, 4.2, 4.3, 4.6, 4.7 Math Nation 10.7 Larson 4.6, 4.6 pt. 2 MAFS.912.S-ID.3.9: Math Nation 10.7 Distinguish between correlation and causation. Larson 4.6, 4.6 pt. 2 8 Mathematical Practice Standards Textbook Section Make sense of problems and persevere in solving them. MAFS.K12.MP.1.1: Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints,
MAFS.K12.MP.2.1: MAFS.K12.MP.3.1: 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. Mathematically proficient 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. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can
MAFS.K12.MP.4.1: MAFS.K12.MP.5.1: construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way 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. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
MAFS.K12.MP.6.1: MAFS.K12.MP.7.1: MAFS.K12.MP.8.1: Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about 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. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well-remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x² + 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 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. Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x² + x + 1), and (x 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.