Secondary Math 1 Curriculum Notebook
Current as of Sept 2018
Standards
Standards indicate the broad goals for a student to master in a course. Standards are typically set by a state or district school board.
Mathematical Practice Standards ............................................................................................................................... Page 4 Unit 1 One Variable Expressions...................................................................................................................................Page 8 Unit 2 Multivariable Expressions...................................................................................................................................Page 8 Unit 3 Functions...................................................................................................................................................................Page 8 Unit 4 Systems......................................................................................................................................................................Page 9 Unit 5 Linear and Exponential Relationships ............................................................................................................ Page 9 Unit 6 Statistics.......................................................................................................................................Page 9 Unit 7 Geometry Definitions and Transformations ................................................................................Page 10 Unit 8 Geometry Congruence .................................................................................................................Page 10 Unit 9 Logic ............................................................................................................................................Page 10 Standards for entire course......................................................................................................................Page 11
Essential Learning Standards
Particular standards/objectives/indicators that a school/district defines as critical for student learning. In fact, they are so critical that students will receive intervention if they are not learned. Essentials are chosen because they 1. have endurance, 2. have leverage, and 3. are important for future learning. Math...................................................................................................................................................................................................Page 12
Curriculum Resources
The materials teachers use to plan, prepare, and deliver instruction, including materials students use to learn about the subject. Such materials include texts, textbooks, tasks, tools, and media. Sometimes organized into a comprehensive program format, they often provide the standards, units, pacing guides, assessments, supplemental resources, interventions, and student materials for a course.
5 Strands of Mathematical Proficiency...............................................................................................................................Page 13 USBE Core Content Guides.......................................................................................................................................................Page 14
Pacing Guide
The order and timeline of the instruction of standards, objectives, indicators, and Essentials over the span of a course (semester or year).
Math Essentials Pacing Guide.................................................................................................................................................Page 89
Units
A plan for several weeks of instruction, usually based on a theme, that includes individual lesson plans. Units often also include: Standards, learning targets/goals, skills, formative and summative assessment, student materials, essential questions, big ideas, vocabulary, questions, and instructional methods. Understanding By Design.........................................................................................................................................................Page 100 Math ................................................................................................................................................................................................... Page
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Assessment Standards
A set of criteria to guide the assessment of student learning in a course that is based on Standards/Essentials of the course; this might include formative assessment practices, summative assessments/practices, common assessment plans, feedback practices, and a schedule for testing.
SAGE ........................................................................................................................................................................................ Page 101 Ethics.......................................................................................................................................................................................Page 102
Intervention Standards
A set of criteria to guide teachers to provide additional instruction to students who did not master the content in Tier 1 instruction. This might include: commercial intervention programs, teacher-developed intervention materials, diagnostic testing, RTI/MTSS processes, and a list of essential knowledge/skills that will prompt intervention if the student does not demonstrate mastery. RTI............................................................................................................................................................................................Page 104 MTSS........................................................................................................................................................................................Page 106
Supplemental Resources
Instructional materials, beyond the main curricular materials, used to strategically fill gaps/weaknesses of the core program materials.
Provo Way Instructional Model...................................................................................................................................Page 108 Mathematics.........................................................................................................................................................................Page 111
Evidence-based Pedagogical Practices
A list of teaching strategies that are supported by adequate, empirical research as being highly effective.
John Hattie ............................................................................................................................................................................ Page 112
Glossary
Terms and acronyms used in this document.........................................................................................................Page 113
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Mathematics
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solutions pathway rather than simply jumping into a solution attempt. The 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 the continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
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2 Reason abstractly and quantitatively
Mathematically proficient students make a sense of the quantities and their relationships in problem situations. Students bring two complimentary 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 the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of tow plausible arguments, distinguish correct logic or reasoning form that which is flawed and–if there is a flaw in an argument–explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
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4 Model with mathematics
Mathematically proficient students can apply mathematics the 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 on 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, possible improving the model if it has not served its purpose.
5 Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
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6 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.
7 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.
8 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.
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SECONDARY MATHEMATICS I
THE FUNDAMENTAL PURPOSE OF SECONDARY MATHEMATICS I is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, or- ganized into units, deepen and extend understanding of linear relationships, in part by con- trasting them with exponential phenomena, and in part by applying linear models to data that exhibit a linear trend. Secondary Mathematics I uses properties and theorems involving congruent gures to deepen and extend understanding of geometric knowledge from prior grades. The nal unit in the course ties together the algebraic and geometric ideas studied. The Mathematical Practice Standards 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.
CRITICAL AREA 1: By the end of eighth grade, students have had a variety of experi- ences working with expressions and creating equations. Students continue this work by using quantities to model and analyze situations, to interpret expressions, and by creating equations to describe situations.
CRITICAL AREA 2: In earlier grades, students de ne, evaluate, and compare functions, and use them to model relationships between quantities. Students will learn function notation and develop the concepts of domain and range. They move beyond view-
ing functions as processes that take inputs and yield outputs, and start viewing func- tions as objects in their own right. 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. They work with functions given by graphs and tables, keeping in mind that, depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas, as well as those that cannot. When functions describe relationships be- tween quantities arising from a context, students reason with the units in which those quantities are measured. 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. They interpret arithmetic sequences as linear functions and geometric se- quences as exponential functions.
CRITICAL AREA 3: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to ana- lyze and solve systems of linear equations in two variables. This area builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. Students develop uency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solu- tion of linear equations and apply related solution techniques and the laws of expo- nents to the creation and solution of simple exponential equations. Students explore
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UTAH CORE STATE STANDARDS for MATHEMATICS
systems of equations and inequalities, and they nd and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them.
CRITICAL AREA 4: This area builds upon students’ prior experiences with data, provid- ing students with more formal means of assessing how a model ts data. Students use regression techniques to describe approximately linear relationships between quanti- ties. They use graphical representations and knowledge of the context to make judg- ments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of t.
CRITICAL AREA 5: In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions (transla- tions, re ections, and rotations) and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congru- ence criteria, based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.
CRITICAL AREA 6: Building on their work with the Pythagorean Theorem in eighth grade to nd distances, students use a rectangular coordinate system to verify geo- metric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.
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UTAH CORE STATE STANDARDS for MATHEMATICS
Strand: MATHEMATICAL PRACTICES (MP)
The Standards for Mathematical Practice in Secondary Mathematics I describe mathematical habits of mind that teachers should seek to develop in their students. Students become math- ematically pro cient in engaging with mathematical content and concepts as they learn, ex- perience, and apply these skills and attitudes (Standards MP.1–8).
Standard SI.MP.1 Make sense of problems and persevere in solving them. Explain the meaning of a problem and look for entry points to its solution. Analyze givens, con- straints, relationships, and goals. Make conjectures about the form and meaning of the solution, plan a solution pathway, and continually monitor progress asking, “Does this make sense?” Consider analogous problems, make connections between multiple repre- sentations, identify the correspondence between di erent approaches, look for trends, and transform algebraic expressions to highlight meaningful mathematics. Check an- swers to problems using a di erent method.
Standard SI.MP.2 Reason abstractly and quantitatively. Make sense of the quantities and their relationships in problem situations. Translate between context and algebraic representations by contextualizing and decontextualizing quantitative relationships. This includes the ability to decontextualize a given situation, representing it algebra- ically and manipulating symbols uently as well as the ability to contextualize algebraic representations to make sense of the problem.
Standard SI.MP.3 Construct viable arguments and critique the reasoning of others. Understand and use stated assumptions, de nitions, and previously established results in constructing arguments. Make conjectures and build a logical progression of state- ments to explore the truth of their conjectures. Justify conclusions and communicate them to others. Respond to the arguments of others by listening, asking clarifying ques- tions, and critiquing the reasoning of others.
Standard SI.MP.4 Model with mathematics. Apply mathematics to solve problems aris- ing in everyday life, society, and the workplace. Make assumptions and approximations, identifying important quantities to construct a mathematical model. Routinely interpret mathematical results in the context of the situation and re ect on whether the results make sense, possibly improving the model if it has not served its purpose.
Standard SI.MP.5 Use appropriate tools strategically. Consider the available tools and be su ciently familiar with them to make sound decisions about when each tool might be helpful, recognizing both the insight to be gained as well as the limitations. Identify relevant external mathematical resources and use them to pose or solve problems. Use tools to explore and deepen their understanding of concepts.
Standard SI.MP.6 Attend to precision. Communicate precisely to others. Use explicit de nitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose. Specify units of measure and label axes to clarify the cor- respondence with quantities in a problem. Calculate accurately and e ciently, express numerical answers with a degree of precision appropriate for the problem context.
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UTAH CORE STATE STANDARDS for MATHEMATICS
Standard SI.MP.7 Look for and make use of structure. Look closely at mathemati-
cal relationships to identify the underlying structure by recognizing a simple structure within a more complicated structure. See complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, 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.
Standard SI.MP.8 Look for and express regularity in repeated reasoning. Notice if rea- soning is repeated, and look for both generalizations and shortcuts. Evaluate the reason- ableness of intermediate results by maintaining oversight of the process while attending to the details.
Strand: NUMBER AND QUANTITY—Quantities (N.Q)
Reason quantitatively and use units to solve problems. Working with quantities and the rela- tionships between them provides grounding for work with expressions, equations, and func- tions (Standards N.Q.1–3).
Standard N.Q.1 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.
Standard N.Q.2 De ne appropriate quantities for the purpose of descriptive modeling. Standard N.Q.3 Choose a level of accuracy appropriate to limitations on measurement
when reporting quantities.
Strand: ALGEBRA—Seeing Structure in Expressions (A.SSE)
Interpret the structure of expressions (Standard A.SSE.1).
Standard A.SSE.1 Interpret linear expressions and exponential expressions with integer exponents that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coe cients.
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.
Strand: ALGEBRA—Creating Equations (A.CED)
Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situa- tions requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).
Standard A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and simple exponential functions.
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Standard A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
Standard A.CED.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 di erent foods.
Standard A.CED.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.
Strand: ALGEBRA—Reasoning With Equations and Inequalities (A.REI)
Understand solving equations as a process of reasoning and explain the reasoning (Standard A.REI.1). Solve equations and inequalities in one variable (Standard A.REI.3). Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. Include cases where the two equations describe the same line—yielding in nitely many solutions—and cases where two equations describe parallel lines—yielding no solution; connect to GPE.5, which requires students to prove the slope criteria for paral- lel lines (Standards A.REI.5–6). Represent and solve equations and inequalities graphically (Standards A.REI.10–12).
Standard A.REI.1 Explain each step in solving a linear 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 solu- tion method. Students will solve exponential equations with logarithms in Secondary Mathematics III.
Standard A.REI.3 Solve equations and inequalities in one variable.
a. Solve one-variable equations and literal equations to highlight a variable of interest.
b. Solve compound inequalities in one variable, including absolute value inequalities.
c. Solve simple exponential equations that rely only on application of the laws of exponents (limit solving exponential equations to those that can be solved without logarithms). For example, 5x = 125 or 2x = 1/16.
Standard A.REI.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.
Standard A.REI.6 Solve systems of linear equations exactly and approximately (numeri- cally, algebraically, graphically), focusing on pairs of linear equations in two variables.
Standard A.REI.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).
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UTAH CORE STATE STANDARDS for MATHEMATICS
Standard A.REI.11 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); nd the solutions approximately; e.g., using technology to graph the functions, make tables of values, or nd successive approximations. Include cases where f(x) and/or g(x) are linear and exponential functions.
Standard A.REI.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 corre- sponding half-planes.
Strand: FUNCTIONS—Interpreting Linear and Exponential Functions (F.IF)
Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examples of linear and exponential func- tions (Standards F.IF.1–3). Interpret linear or exponential functions that arise in applications in terms of a context (Standards F.IF.4–6). Analyze linear or exponential functions using dif- ferent representations (Standards F.IF.7,9).
Standard F.IF.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).
Standard F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Standard F.IF.3 Recognize that sequences are functions, sometimes de ned recursively, whose domain is a subset of the integers. Emphasize arithmetic and geometric sequenc- es as examples of linear and exponential functions. For example, the Fibonacci sequence is de ned recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Standard F.IF.4 For a function that models a relationship between two quantities, in- terpret 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; rela- tive maximums and minimums; symmetries; and end behavior.
Standard F.IF.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.
Standard F.IF.6 Calculate and interpret the average rate of change of a function (present- ed symbolically or as a table) over a speci ed interval. Estimate the rate of change from a graph.
Standard F.IF.7 Graph functions expressed symbolically and show key features of the
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graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear functions and show intercepts.
e. Graph exponential functions, showing intercepts and end behavior.
Standard F.IF.9 Compare properties of two functions, each represented in a di erent way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, compare the growth of two linear functions, or two exponential functions such as y=3n and y=100•2n.
Strand: FUNCTIONS—Building Linear or Exponential Functions (F.BF)
Build a linear or exponential function that models a relationship between two quantities (Standards F.BF.1–2). Build new functions from existing functions (Standard F.BF.3).
Standard F.BF.1 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.
Standard F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Limit to linear and exponential functions. Connect arithmetic sequences to linear func- tions and geometric sequences to exponential functions.
Standard F.BF.3 Identify the e ect on the graph of replacing f(x) by f(x) + k, for speci c values of k (both positive and negative); nd the value of k given the graphs. Relate the vertical translation of a linear function to its y-intercept. Experiment with cases and illus- trate an explanation of the e ects on the graph using technology.
Strand: FUNCTIONS—Linear and Exponential (F.LE)
Construct and compare linear and exponential models and solve problems (Standards F.LE.1–
3). Interpret expressions for functions in terms of the situation they model. (Standard F.LE.5). Standard F.LE.1 Distinguish between situations that can be modeled with linear functions
and with exponential functions.
a. Prove that linear functions grow by equal di erences over equal intervals; exponen- tial functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit inter- val relative to another.
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UTAH CORE STATE STANDARDS for MATHEMATICS
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Standard F.LE.2 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).
Standard F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly.
Standard F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. Limit exponential functions to those of the form f(x) = bx + k.
Strand: GEOMETRY—Congruence (G.CO)
Experiment with transformations in the plane. Build on student experience with rigid mo- tions from earlier grades (Standards G.CO.1–5). Understand congruence in terms of rigid mo- tions. Rigid motions are at the foundation of the de nition of congruence. Reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems (Standards G.CO.6–8). Make geometric constructions (Standards G.CO.12–13).
Standard G.CO.1 Know precise de nitions of angle, circle, perpendicular line, parallel line, and line segment, based on the unde ned notions of point, line, distance along a line, and distance around a circular arc.
Standard G.CO.2 Represent transformations in the plane using, for example, transparen- cies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that pre- serve distance and angle to those that do not (e.g., translation versus horizontal stretch).
Standard G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and re ections that carry it onto itself.
Standard G.CO.4 Develop de nitions of rotations, re ections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Standard G.CO.5 Given a geometric gure and a rotation, re ection, or translation, draw the transformed gure using, for example, graph paper, tracing paper, or geometry soft- ware. Specify a sequence of transformations that will carry a given gure onto another. Point out the basis of rigid motions in geometric concepts, for example, translations move points a speci ed distance along a line parallel to a speci ed line; rotations move objects along a circular arc with a speci ed center through a speci ed angle.
Standard G.CO.6 Use geometric descriptions of rigid motions to transform gures and to predict the e ect of a given rigid motion on a given gure; given two gures, use
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the de nition of congruence in terms of rigid motions to decide whether they are congruent.
Standard G.CO.7 Use the de nition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and correspond- ing pairs of angles are congruent.
Standard G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) fol- low from the de nition of congruence in terms of rigid motions.
Standard G.CO.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, re ective devices, paper folding, dynamic geometric software, etc.). Emphasize the ability to formalize and defend how these constructions result in the desired objects. For example, copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Standard G.CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Emphasize the ability to formalize and defend how these construc- tions result in the desired objects.
Strand: GEOMETRY—Expressing Geometric Properties With Equations (G.GPE)
Use coordinates to prove simple geometric theorems algebraically (Standards G.GPE.4–5, 7).
Standard G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a gure de ned by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Standard G.GPE.5 Prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems (e.g., nd the equation of a line parallel or perpendicular to a given line that passes through a given point).
Standard G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles; connect with The Pythagorean Theorem and the distance formula.
Strand: STATISTICS AND PROBABILITY—
Interpreting Categorical and Quantitative Data (S.ID)
Summarize, represent, and interpret data on a single count or measurement variable (Standards S.ID.1–3). Summarize, represent, and interpret data on two categorical and quan- titative variables (Standard S.ID.6). Interpret linear models building on students’ work with linear relationships, and introduce the correlation coe cient (Standards S.ID.7–9).
UTAH CORE STATE STANDARDS for MATHEMATICS
SECONDARY MATHEMATICS I | 11
UTAH CORE STATE STANDARDS for MATHEMATICS
Standard S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
Standard S.ID.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 di erent data sets.
Standard S.ID.3 Interpret di erences in shape, center, and spread in the context of the data sets, accounting for possible e ects of extreme data points (outliers). Calculate the weighted average of a distribution and interpret it as a measure of center.
Standard S.ID.6 Represent data on two quantitative variables on a scatter plot, and de- scribe how the variables are related.
a. Fit a linear function to the data; use functions tted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the con- text. Emphasize linear and exponential models.
b. Informally assess the t of a function by plotting and analyzing residuals. Focus on situations for which linear models are appropriate.
c. Fit a linear function for scatter plots that suggest a linear association.
Standard S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.
Standard S.ID.8 Compute (using technology) and interpret the correlation coe cient of a linear t.
Standard S.ID.9 Distinguish between correlation and causation.
SECONDARY MATHEMATICS I | 12
Secondary Math 1: Essentials and Good---to---Know Standards
Essential Standards
1.A.SSE.1a: interpret parts of an expression (terms, factors, coefficients) 1.A.CED.1: create and solve equations in one variable (linear) 1.A.CED.2: create equations in two or more variables –graph (linear)
1.A.CED.1: create and solve equations and inequalities in one variable (linear and exponential)
1.A.CED.2: create equations in two or more variables –graph (linear and exponential) 1.A.CED.3: represent constraints
by equations or inequalities; interpret solutions as viable or non-viable
1.A.CED.3: represent constraints by equations or inequalities; interpret solutions as viable or non-viable
1. A.CED.4: rearrange formulas to highlight quantity of interest
1.A.REI.1: explain each step in solving a linear equation
1.A.REI.3a: solve one-variable equations and literal equations to highlight a variable of interest
1.A.REI.5: prove that, given a system of two equations in two variables, replacing one equation with a sum of that
equation and a multiple of the other produces a system with the same solutions
1.A.REI.6: solve systems of linear equations exactly and approximately (numerically, algebraically, graphically) 1.A.REI.10: understand that a graph of an equation in two variables is the set of all solutions plotted in the coordinate
plane (linear and exponential)
1.A.REI.11: explain why the x-coordinates of the points where the graphs of the equations intersect are the solutions 1.A.REI.12: graph the solutions to a linear inequality
1.F.IF.2: 1.F.IF.3:
1.F.IF.6: 1.F.IF.7a: 1.F.IF.7e: 1.F.IF.9:
1.F.LE.1c:
1.F.LE.2:
1.F.LE.5:
1.S.ID.1: 1.S.ID.2:
1.S.ID.3:
1.S.ID.6a:
1.S.ID.7: 1.G.CO.5:
1.G.CO.6: 1.G.CO.7: 1.G.CO.8:
use function notation
understand that sequences are functions, sometimes defined recursively, whose domain is a subset of integers (arithmetic/linear and geometric/exponential)
calculate and interpret the average rate of change of a function over a specified interval; estimate from a graph graph linear functions and show intercepts
graph exponential functions, showing intercepts and end behavior
compare properties of 2 linear functions, 2 exponential functions, or linear vs exponential, each represented in a different way (algebraically, graphically, numerically in tables or by verbal descriptions)
recognize situations in which a quantity grows or decays by a constant percent rate per unit interval
relative to another
construct linear or exponential functions given a graph, description of a relationship or two input-output
pairs
interpret the parameters in a linear function in terms of a context (slope, y- intercept) and interpret the parameters in an exponential function in terms of a context (base value, vertical shifts)
represent data with plots on the real number line (dot plots, histograms, and box plots)
use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (IQR, range and standard deviation) of two or more data sets
interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers); calculate the weighted average of a distribution and interpret it as a measure of center
represent data on two quantitative variables on a scatter plot and describe how they are related; fit a linear function to the data; use fitted functions to data to solve problems in the context of the data given; emphasize linear and exponential models
interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of data given a geometric figure and a rotation, reflection or translation, draw the transformed figure using a variety of media; specify a sequence of transformations that will carry a given figure onto another
use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of rigid motion to decide if they are congruent use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and angles are congruent
explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence
in terms of rigid motions
1.G.GPE.4: use coordinates to prove simple geometric theorems algebraically focusing on lines, segments and angles 1.G.GPE.5: prove the slope criteria for parallel and perpendicular lines; use them to solve geometric problems 1.G.GPE.7: use coordinates to compute perimeters of polygons and areas of triangles and rectangles, using the distance
formula
Good to know Standards
1.A.SSE.1b: interpret complicated expressions by viewing one or more of their parts as a single entity
1.F.IF.1:
1.F.IF.4:
understand that a function from one set (domain) to another set (range) assigns to each element of the domain exactly one element of the range.
interpret key features of linear graphs (intercepts, increasing/decreasing, +/-, max/min, symmetries, end
behavior)
relate the domain of a function to its graph
1.F.IF.5:
1.F.LE.1a: prove linear functions grow by equal differences over equal intervals 1.F.LE.1b: recognize situations in
which one quantity changes at a constant rate per unit interval relative to another
1.A.REI.3b: solve compound inequalities, including absolute value inequalities
1.A.REI.3c: solve simple exponential equations
1.F.IF.4: interpret key features of linear and exponential graphs (intercepts, increasing/decreasing, +/-, max/min,
symmetries, end behavior)
1.F.BF.1a: write a function that describes a relationship between two quantities: determine an explicit expression, a
recursive process, or steps for calculation from a context
1.F.BF.1b: write a function that describes a relationship between two quantities: combine standard function types
1.F.BF.2: 1.F.BF.3:
1.S.ID.6b: 1.S.ID.6c: 1.S.ID.8: 1.S.ID.9: 1.G.CO.1:
using arithmetic operations
write arithmetic (linear) and geometric (exponential) sequences both recursively and with an explicit
formula
identify the effect on the graph of replacing f(x) by f(x)+k (vertical translation) 1.F.LE.1a: prove exponential functions grow by equal factors over equal intervals 1.F.LE.3: observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly
informally assess the fit of a function by plotting and analyzing residuals (linear)
fit a linear function for scatter plots that suggest a linear association
compute (using technology) and interpret the correlation coefficient of a linear fit
distinguish between correlation and causation
know precise definitions of angle, circle perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc
1.G.CO.2: represent transformations in a plane using a variety of media; describe transformations as functions that take points in a plane as inputs and gives other points as outputs; compare transformations that preserve
1.G.CO.3:
1.G.CO.4:
distance and angle and those that do not
given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry
it onto itself
develop definitions of rotations, reflections and translations in terms of angles, circles, perpendicular lines,
parallel lines and line segments
1.G.CO.12: make formal geometric constructions with a variety of tools and methods; copying a segment, copying an
angle, bisecting a segment, bisecting an angle, constructing perpendicular lines, including the perpendicular bisector of a line segment, and constructing a line parallel to a given line through a point not on a line
1.G.CO.13: construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle
Curriculum Resources
5 Strands of Mathematical Proficiency from NRC’s Adding It Up
Conceptual understanding: Comprehension of mathematical concepts, operations, and relations
Procedural fluency: skill in carrying out procedures flexibly, accurately, efficiently and appropriately
Strategic competence: ability to formulate, represent, and solve mathematical problems
Adaptive reasoning: capacity for logical thought, reflection, explanation, and justification
Productive disposition: habitual inclination to see mathematics as sensible, useful, worthwhile, coupled with a belief in diligence and one's own efficacy
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
13
Secondary Math 1 State Core Guides
Number and Quantity (NQ) .............................................................................................. Page 22 Algebra – Seeing Structure in Expressions (ASSE) ............................................................. Page 25 Algebra – Creating Equations (ACED) ................................................................................ Page 26 Algebra – Reasoning with Equations and Inequalities (AREI)............................................ Page 30 Functions – Interpreting Linear and Exponential Functions (FIF)...................................... Page 37 Functions – Build New Functions from Existing Functions (FBF)....................................... Page 45 Functions – Linear and Exponential (FLE).......................................................................... Page 48 Geometry – Congruence (GCO)......................................................................................... Page 51 Geometry – Expressing Geometric Properties with Equations (GPE) ............................... Page 61 Statistics – Interpreting Categorical and Quantitative Data (SID) ..................................... Page 64
Support for Teachers
Number and Quantity Core Guide
Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions (Standards N.Q.1–3)
StandardN.Q.1: Useunitsasawaytounderstandproblemsandtoguidethesolutionofmulti-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
Concepts and Skills to Master
• Use units as a way to understand problems and to guide the solution of multi-step problems.
• Chooseand interpretunitsconsistentlyinformulas.
• Choose and interpret the scale and the origin in graphs and data displays.
Related Standards: Current Course
Related Standards: Future Courses
I.N.Q.2, I.N.Q.3, I.A.CED.2, I.ACED.4, I.AREI.10, I.A.REI.11, I.A.REI.12,
I.F.IF.5, I.F.IF.6, I.F.IF.7, I.F.LE.1, I.F.LE.3, I.S.ID.1, I.S.ID.6, I.S.ID.7
All standards related to expressions, equations, functions, and data displays
Critical Background Knowledge (Access Background Knowledge)
• Graph points in all four quadrants of the coordinate plane (6.NS.8)
• Choose appropriate graph/plot for data (6.SP .4)
• Compute unit rates involving ratios of lengths, areas, and other quantities (7.RP .1)
• Approximately locate irrational numbers on a number line diagram (8.NS.2)
• Analyzefeaturesofagraphandsketchgraphsthathavebeendescribedverbally(8.F.5)
• Construct and interpret scatter plots (8.SP .1)
Academic Vocabulary
Scale, units of measurement
Resources:
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70106
1.NQ.1
Support for Teachers
Number and Quantity Core Guide
Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions (Standards N.Q.1–3)
Standard N.Q.2: Define appropriate quantities for the purpose of descriptive modeling.
Concepts and Skills to Master
• Choose appropriate measures and units when creating a model for data (descriptive modeling).
Related Standards: Current Course
Related Standards: Future Courses
I.N.Q.1, I.N.Q.3, I.A.CED.3, I.F.IF.4, I.S.ID.6
III.G.MG.1, III.G.MG.2, III.G.MG.3
Critical Background Knowledge (Access Background Knowledge)
• Choose appropriate graph/plot for data (6.SP .4)
• Construct and interpret scatter plots (8.SP .1)
Academic Vocabulary
Descriptive modeling
Resources:
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70106
1.N.Q.2
Support for Teachers
Number and Quantity Core Guide
Reason quantitatively and use units to solve problems. Working with quantities and the relationships between them provides grounding for work with expressions, equations, and functions (Standards N.Q.1–3)
Standard N.Q.3: Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Concepts and Skills to Master
• Determine whether whole numbers, fractions, or decimals are most appropriate.
• Determine the appropriate power of ten to reasonably measure a quantity.
• Determine the resulting accuracy in calculations.
• Determine what level of rounding should be used in a problem situation.
Related Standards: Current Course
Related Standards: Future Courses
I.N.Q.1, I.N.Q.2, I.SI.MP.6, all standards related to expressions, equations, and functions
II.SI.MP.6, III.SI.MP.6, all standards related to expressions, equations, and functions
Critical Background Knowledge (Access Background Knowledge)
• Know relative sizes of measurement units (4.MD.1)
• Approximately locate irrational numbers on a number line diagram (8.NS.2)
• Use powers of ten to estimate very large or very small quantities (8.EE.3)
• Attend to precision (I.SI.MP .6)
Academic Vocabulary
Precision, accuracy
Resources:
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70106
1.N.Q.3
Algebra – Seeing Structure in Expressions Core Guide
Interpret the structure of expressions (Standard A.SSE.1)
Standard 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.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example,
interpret P(1+r)nastheproductofPandafactornotdependingonP.
Concepts and Skills to Master
• Given an expression, identify the terms, base, exponents, coefficients, and factors.
• Determine the real world context of the variables in an expression.
• Explain the context of different parts of a formula.
Related Standards: Current Course
Related Standards: Future Courses
I.SI.MP.7, I.A.CED.2, I.A.CED.4, I.F.IF.3, I.F.BF.2, I.F.BF.3, I.F.LE.1, I.S.ID.7
II.N.RN.1, II.N.RN.2, II.A.SSE.1, II.A.SSE.2, II.A.SSE.3, II.A.CED.2,
II.A.CED.4, II.A.REI.4, II.F.IF.8, II.F.BF.3, III.A.SSE.1, III.A.SSE.2,
III.A.SSE.4, III.A.CED.2, II.A.CED.4, III.A.APR.3, III.A.APR.5, III.F.IF.8, III.F.BF.3, III.F.LE.4, III.F.TF.5
Support for Teachers
Critical Background Knowledge (Access Background Knowledge)
• Understand that rewriting an expression can highlight quantities (7.EE.2)
• Determine rate of change and initial value of a function (8.F.3, 8.F.4)
• Interpret unit rate as the slope (8.EE.5)
Academic Vocabulary
Exponents, factors, terms, bases, coefficients, expression
Resources:
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70114
1.A.SSE.1
Creating Equations Core Guide
Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).
Standard I.A.CED.1: Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and simple exponential functions.
Concepts and Skills to Master
• Create linear and exponential equations and inequalities in one variable and use them to solve problems.
• Show solutions to inequalities using set notation, interval notation, and inequalities.
• Use properties of exponents to solve exponential equations and inequalities, limit to situations to integer
solutions (solving more complicated exponential functions using Logarithms occurs in Secondary Mathematics III).
Related Standards: Current Course
Related Standards: Future Courses
I.A.REI.1, I.A.REI.3
II.A.CED.1, III.A.CED.1, II.A.REI.4
Support for Teachers
Critical Background Knowledge
• Create expressions and equations (6.EE.6 and 7.EE.4)
• Solve equations (6.EE.7, 7.EE.4a, and 8.EE.7)
• Solve inequalities and use inequality notation (6.EE.8, 7.EE.4b, and 8.EE.7)
• Understand and use properties of exponents (8.EE.1)
Academic Vocabulary
Exponential equations, set notation, interval notation
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70123
I.A.CED.1
Creating Equations Core Guide
Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).
StandardI.A.CED.2:Createequationsintwoormorevariablestorepresentrelationships betweenquantities;graph equations on coordinate axes with labels and scales.
Concepts and Skills to Master
• Create and graph an equation to represent a linear or exponential relationship between two quantities.
• Create linear and exponential equations from various models.
• Graph equations on coordinate axes with appropriate labels and scales.
Related Standards: Current Course
Related Standards: Future Courses
I.N.Q.1, I.N.Q.2, I.A.CED.1, I.A.SSE.1, I.A.SSE.2, I.A.SSE.3, I.A.REI.6,
I.A.REI.10, I.F.IF.4, I.F.IF.5, I.F.IF.7, I.F.BF.1, I.F.BF.2, I.F.BF.3, I.F.LE.3
II.A.CED.1, II.A.SSE.1, II.A.SSE.2, II.A.SSE.3, II.F.IF.4, II.F.IF.5, II.F.IF.7,
II.F.BF.1, II.F.BF.3, II.F.LE.3, III.A.CED.1, III.A.SSE.1, III.A.SSE.2,
III.A.SSE.4, III.F.IF.4, III.F.IF.5, III.F.IF.7, III.F.BF.1, III.F.BF.3, III.F.LE.3
Support for Teachers
Critical Background Knowledge
• Create expressions and equations in two variables (6.EE.9 and 7.EE.4)
• Construct a function to model a linear relationship between two quantities (8.F.4)
• Describequalitativelytherelationshipbetweentwoquantitiesbyanalyzingagraph(8.F.5)
Academic Vocabulary
dependent variable, independent variable, scale
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70123
I.A.CED.2
Creating Equations Core Guide
Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).
StandardI.A.CED.3:Representconstraintsbyequationsorinequalitiesandbysystemsof equationsand/or inequalities,andinterpret solutionsasviableornon-viableoptions in a modeling context. Forexample,represent inequalities describing nutritional and cost constraints on combinations of different foods.
Concepts and Skills to Master
• Write and graph equations and inequalities representing constraints in contextual situations. • Determinewhetherapointisasolutiontoanequationorinequality.
• Interpret the meaning and viability of a solution based on the context.
Related Standards: Current Course
Related Standards: Future Courses
I.N.Q.2, I.A.CED.1, I.A.CED.2, I.A.CED.4, I.A.REI.5, I.A.REI.6, I.A.REI.10,
I.A.REI.11, I.A.REI.12, I.F.IF.4, I.F.IF.5, I.F.IF.7, I.S.ID.7
II.A.CED.1, II.A.CED.2, II.A.CED.4, II.A.REI.7, II.F.IF.4, II.F.IF.5, II.F.IF.7,
III.A.CED.1, III.A.CED.2, III.A.CED.3, III.A.CED.4, III.A.REI.2, III.A.REI.11, III.F.IF.4, III.F.IF.5, III.F.IF.7, P.N.VM.13
Support for Teachers
Critical Background Knowledge
• Reason about and solve one-variable equations and inequalities (6.EE.5 – 8)
• Solve word problems leading to equations and inequalities (in one variable) (7.EE.4 and 8.EE.7)
• Analyze and solve systems of linear equations graphically (8.EE.8)
Academic Vocabulary
Constraint, viable, half-plane, solution region
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70123
I.A.CED.3
Creating Equations Core Guide
Create equations that describe numbers or relationships. Limit these to linear equations and inequalities, and exponential equations. In the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs (Standards A.CED.1–4).
StandardI.A.CED.4:Rearrangeformulastohighlightaquantityofinterest,usingthesame reasoningasinsolving equations. Forexample, rearrangeOhm’sLawV=IRtohighlight resistanceR.
Concepts and Skills to Master
• Extend the concepts used in solving numerical equations to rearranging formulas for a particular variable; limit to linear formulas
Related Standards: Current Course
Related Standards: Future Courses
I.A.REI.3, I.A.REI.5, I.A.CED.1, I.A.CED.2, I.A.SSE.1
II.A.SSE.1, II.A.CED.1, II.A.CED.2, II.A.CED.4, II.F.IF.8, II.G.GMD.1, II.G.GMD.3, III.A.CED.4, III.F.IF.8, III.A.SSE.1
Support for Teachers
Critical Background Knowledge
• Use variables to write equations and solve them (6.EE.2, 6.EE.6, and 6.EE.7)
• Understand that rewriting an expression in different forms in a problem context can shed light on the
problem and how quantities are related (7.EE.2)
• Solve multi-step equations (7.EE.3, 7.EE.4, and 8.EE.7)
Academic Vocabulary
Constant, variable, formula, literal equation
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70123
I.A.CED.4
Reasoning with Equations and Inequalities Core Guide
Understand solving equations as a process of reasoning and explain the reasoning (Standard A.REI.1)
Standard I.A.REI.1: Explain each step in solving a linear 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. Students will solve exponential equations with logarithms in
SC e o c n o c n e d p a t s r y a n M d a S t h k e i l ml s a t o t i c Ms I a I I s . t e r
•Understand, apply, and explain the results of using inverse operations.
•Justify the steps in solving equations by applying and explaining the properties of equality, inverse, and
identity.
Related Standards: Current Course
Related Standards: Future Courses
I.A.CED.1, I.A.CED.4, I.A.REI.3
II.N.CN.2, II.N.CN.8, II.A.SSE.2, II.A.SSE.3, II.A.APR.1, II.A.CED.1,
II.A.CED.4, II.F.TF.8, III.A.APR.1, III.A.APR.7, III.A.CED.1, III.A.CED.4,
III.A.REI.2, III.F.IF.8, P.N.VM.8, P.N.VM.9, P.N.VM.10, and
all standards
Support for Teachers related to proof and inverse
Critical Background Knowledge (Access Background Knowledge)
• Use properties of operations (throughout elementary: OA standards) to create equivalent expressions (7.EE.1 and 7.EE.2)
• Apply and extend previous understandings of arithmetic to algebraic expressions (6.EE.3, 6.EE.4)
• Reason about and solve one variable equations (6.EE.5, 6.EE.6, 6.EE.7)
Ac•adeSmoiclvVeoocnaebuvlaryiable equations (7.EE.4a and 8.EE.7)
Properties of operations (associative, commutative, distributive) and properties of equality (reflexive, transitive, symmetric), properties of inverse (multiplicative inverse)
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70134
I.A.REI.1
Reasoning with Equations and Inequalities Core Guide
Solve equations and inequalities in one variable (Standard A.REI.3)
Standard I.A.REI.3: Solve equations and inequalities in one variable.
a. Solve one-variable equations and literal equations to highlight a variable of interest.
b. Solve compound inequalities in one variable, including absolute value inequalities.
c. Solve simple exponential equations that rely only on application of the laws of exponents (limit solving
exponential equations to thosethatcanbesolvedwithout logarithms).Forexample,5 =125or2 =
xx
Concepts and Skills to Master
• Solve one-variable equations and literal equations to highlight a variable of interest.
• Understand and apply the properties of compound inequalities.
• Solve compound inequalities in one variable.
• Solve simple exponential equations that rely only on application of the laws of exponents
(limit solving exponential equations to those that can be solved without logarithms).
• Solve absolute value inequalities.
Related Standards: Current Course
Related Standards: Future Courses
I.A.CED.1, I.A.CED.4, I.A.SSE.1, I.A.REI.1, plus all two variable equations (rest of algebra and
All algebra standards (algebra and function is used throughout high school mathematics courses), all F.BF
1/16.
function standards) standards Support for Teachers
Critical Background Knowledge
• Solving linear equations and inequalities in one variable (8.EE.7)
• Solve one- step equations (6.EE.7)and inequalities (6.EE.8)
• Solve two-step equations and inequalities (7.EE.4)
• Understand absolute value (6.NS.7) and solve absolute equations (8.EE.7c)
• Reason about and solve equations and inequalities (6.EE.5-6)
• Use properties of algebra to simplify algebraic expressions (7.EE.1-2)
Academic Vocabulary
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70134
I.A.REI.3
Reasoning with Equations and Inequalities Core Guide
Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. 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, which requires students to prove the slope criteria for parallel lines (Standard A.REI.5-6)
Standard I.A.REI.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.
Concepts and Skills to Master
• Explain the use of the multiplication property of equality to solve a system of equations.
• Explain why the sum of two equations is justifiable in the solving of a system of equations. (Property of
equality)
• Relate the process of linear combinations with the process of substitution for solving a system of linear
equations.
Related Standards: Current Course
Related Standards: Future Courses
I.A.SSE.1, I.A.CED.2, I.A.CED.3, I.A.REI.6, I.A.REI.10, I.A.REI.11, I.A.REI.12
II.A.REI.7, all systems of equations
Support for Teachers
Critical Background Knowledge
• Graph a line (8.F.4)
• Solve systems of equations graphically (8.EE.8)
Academic Vocabulary
Elimination by multiplication and addition, substitution
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70134
I.A.REI.5
Reasoning with Equations and Inequalities Core Guide
Solve systems of equations. Build on student experiences graphing and solving systems of linear equations from middle school. 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, which requires students to prove the slope criteria for parallel lines (Standard A.REI.5-6)
Standard I.A.REI.6: Solve systems of linear equations exactly and approximately (numerically, algebraically, graphically), focusing on pairs of linear equations in two variables.
Concepts and Skills to Master
• Solve a system of linear equations using various representations (numerically, algebraically, and graphically).
• •Use structure to predict one, infinitely many or no solutions.
Related Standards: Current Course
Related Standards: Future Courses
I.A.REI.3, I.A.REI.5, I.A.REI.10, I.A.REI.11, I.A.REI.12, I.A.CED.2, I.A.CED.3, I.A.CED.4, I.F.IF.9, I.F.BF.1, I.G.GPE.4, I.G.GPE.5
II.A.REI.7, II.A.APR.1, II.A.CED.1, II.A.CED.2, II.A.CED.4, II.A.SSE.2,II.A.SSE.3, II.F.LE.3; III.A.CED.2, III.A.CED.3, III.A.REI.11, P.N.VM.6,P.N.VM.7, P.N.VM.8, P.N.VM.13, P.A.REI.8, P.A.REI.9
Support for Teachers
Critical Background Knowledge
• Solve systems of equations graphically (8.EE.8)
• Use properties of operations to generate equivalent expressions (7.EE.1)
Academic Vocabulary
System of equations, consistent and inconsistent systems, dependent and independent systems, solution set
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70134
I.A.REI.6
Reasoning with Equations and Inequalities Core Guide
Represent and solve equations and inequalities graphically (Standards A.REI.10-12)
Standard I.A.REI.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).
Concepts and Skills to Master
• Identify solutions and non-solutions of linear and exponential equations.
• Graph points that satisfy linear and exponential equations.
• Understand that a continuous curve or a line contains an infinite number of solutions.
Related Standards: Current Course
Related Standards: Future Courses
I.A.CED.2, I.A.REI.11, I.A.REI.12, I.F.IF.1, I.F.IF.2, I.F.IF.4, I.F.IF.5, I.F.IF.7, I.F.BF.1, I.F.LE.2, I.S.ID.6
II.F.IF.4, II.F.IF.5, II.F.IF.7, II.F.BF.1, III.F.IF.4, III.F.IF.5, III.F.IF.7, III.F.TF.2, P.F.IF.7
Support for Teachers
Critical Background Knowledge
• Solve mathematical problems by graphing points in all four quadrants (6.NS.8)
• Understand that solutions to equations are values that make the equation or inequality true (6.EE.5)
• Understand that a graph of a function is the set of ordered pairs consisting of an input and a corresponding output
(8.F.1)
• Construct a function to model a relationship between two quantities (8.F.4)
Academic Vocabulary
Ordered pair, coordinate plane, solution, non-solution, sets
Resources
Curriculum Resources: https://www.uen.org/core/core.do?courseNum=5600#70136
I.A.REI.10
Reasoning with Equations and Inequalities Core Guide
Represent and solve equations and inequalities graphically (Standards A.REI.10-12)
StandardI.A.REI.11:Explainwhythex-coordinatesofthepointswherethegraphsoftheequationsy=f(x)andy= g(x)intersectarethe solutionsoftheequationf(x)=g(x);findthesolutionsapproximately,e.g.,usingtechnologyto graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear and exponential functions. ★
Concepts and Skills to Master
• Approximate solutions to systems of two equations using graphing technology.
• Approximate solutions to systems of two equations using tables of values.
• 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).
• Be able to express that when f(x) = g(x), the two equations have the same solution(s).
Related Standards: Current Course
Related Standards: Future Courses
I.A.REI.5, I.A.REI.6, I.A.REI.10, I.A.REI.12, I.F.IF.1, I.F.IF.2, I.F.IF.9, I.F.LE.3
II.F.IF.9, II.F.LE.3, III.A.REI.11, III.F.IF.9, III.F.LE.3, P.VM.13
Support for Teachers
Critical Background Knowledge
• Understand that solutions to equations are values that make the equation true (6.EE.5)
• Understand that a graph of a function is the set of ordered pairs consisting of an input and a corresponding output
(8.F.1)
• Give examples of linear equations with one solution, infinitely many solutions or no solutions (8.EE.7a)
• Understand that a solution to a system of two linear equations corresponds to points of intersection of their graphs
(8.EE.8a)
Academic Vocabulary
Function, intersection, approximate, linear, exponential, f(x), g(x)
Resources
Curriculum Resources: https://www.uen.org/core/core.do?courseNum=5600#70137
I.A.REI.11
Reasoning with Equations and Inequalities Core Guide
Represent and solve equations and inequalities graphically (Standards A.REI.10-12)
Standard I.A.REI.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.
Concepts and Skills to Master
• Graph the solution to linear inequalities in two variables.
• Graph the solution to systems of linear inequalities in two variables.
• Identify the solutions as a region of the plane.
Related Standards: Current Course
Related Standards: Future Courses
I.A.CED.3, I.A.REI.5, I.A.REI.6, I.A.REI.10, I.A.REI.11, I.F.IF.1, I.F.IF.2, I.F.IF.9, I.F.LE.3
III.A.CED.3
Support for Teachers
Critical Background Knowledge
• Understand and solve simple one-step (6.EE.5), two-step (7.EE.4b) and multi-step (8.EE.7b) inequalities in one variable
• Understand that solutions are values that make the inequality true (6.EE.5)
• Recognize that there may be infinitely many solutions to an inequality (6.EE.8)
• Understand that a graph of a function is the set of ordered pairs consisting of an input and a corresponding
output (8.F.1)
• Understand that a solution to a system of two linear equations corresponds to points of intersection of their graphs (8.EE.8a)
Academic Vocabulary
One variable inequality, two variable inequality, half-plane, solution region
Resources
Curriculum Resources: https://www.uen.org/core/core.do?courseNum=5600#70222
I.A.REI.12
Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examplesoflinearandexponentialfunctions(F.IF.1-3)
Standard I.F.IF.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).
Concepts and Skills to Master
• Understand the definition of a function in terms of mapping elements from one set (domain) to another set (range).
• Explain how a given representation of a function (graph, table, equation, context, geometric model) can be used to identify elements of the domain and corresponding elements of the range (x, f(x)).
• Understand the graph of f is the graph of the equation y=f(x).
Related Standards: Current Course
Related Standards: Future Courses
All function standards (functions are used throughout high school mathematicscourses),I.A.REI.10,I.F.IF.5
All function standards (functions are used throughout high school mathematics courses)
Support for Teachers
Critical Background Knowledge
• A function is a rule that assigns to each input exactly one output (8.F.1)
• Multiplerepresentations(tables,graphs,equations,context,geometricmodels)(8.F.2)
Academic Vocabulary
Domain, range, function, input, output, corresponding, set, element
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examplesoflinearandexponentialfunctions(F.IF.1-3)
Standard I.F.IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
Concepts and Skills to Master
• Use function notation
• Evaluate functions, including functions created using arithmetic operations (example: f(x) + g(x) or f(x) – g(x)).
• Interpret statements that use function notation in terms of a context (example: given a context, explain f(5) =
12)
Related Standards: Current Course
Related Standards: Future Courses
All function standards (function notation is used throughout high school mathematics courses)
All function standards (function notation is used throughout high school mathematics courses)
Support for Teachers
Critical Background Knowledge
• Evaluate expressions (6.EE.2c)
Academic Vocabulary
Function notation, evaluate, input, domain, output, range
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Understand the concept of a linear or exponential function and use function notation. Recognize arithmetic and geometric sequences as examplesoflinearandexponentialfunctions(F.IF.1-3)
StandardI.F.IF.3:Recognizethatsequencesarefunctions,sometimesdefinedrecursively, whosedomainisa subset of the integers. Recognize arithmetic and geometric sequences as examples of linear and exponential functions. For example, the Fibonacci sequence is definedrecursivelybyf(0)=f(1)=1,f(n+1)=f(n)+f(n-1)forn≥1.
Concepts and Skills to Master
• Recognize that sequences are functions (recognize the domain is the number of the term and the range is the value of the term).
• Define and express a recursive sequence as a function.
• Recognize that a sequence has a domain which is a subset of integers.
Related Standards: Current Course
Related Standards: Future Courses
I.F.BF.1a, I.F.BF.2, I.F.LE.1, I.F.LE.2
II.F.BF.1a, III.A.SSE.4
Support for Teachers
Critical Background Knowledge
• Use function notation (I.F.IF.2)
• Understanddefinitionoffunction(8.F.1andI.F.IF.1)
• Recognize sequences (taught concurrently with I.F.BF.1, I.F.BF.2)
Academic Vocabulary
Recursive, sequence, functions, domain, range, subset, term
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Interpret linear or exponential functions that arise in applications in terms of a context (F.IF.4-6)
Standard I.F.IF.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 maximums and minimums; symmetries; and end behavior.
Concepts and Skills to Master
• Given a graph, identify key features including x- and y-intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
• •Given a table of values, identify key features such as x- and y-intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; and end behavior.
• Use key features to sketch a graph of the function.
• Use interval notation and symbols of inequality to communicate key features of graphs.
Related Standards: Current Course
Related Standards: Future Courses
I.F.IF.6, I.F.IF.7, F.IF.9, I.F.LE.1, I.F.LE.3
II.F.IF.4, II.F.IF.6, II.F.IF.7, II.F.IF.9, II.F.LE.3, III.F.IF.4, III.F.IF.6, III.F.IF.7
Support for Teachers
Critical Background Knowledge
• Ability to graph a linear (8.F.2) or exponential function from a table or equation
Academic Vocabulary
Increasing, decreasing, positive, negative, intervals, intercepts, interval notation, maximum, minimum, symmetry, and end behavior
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Interpret linear or exponential functions that arise in applications in terms of a context (F.IF.4-6)
Standard I.F.IF.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.
Concepts and Skills to Master
• Identify domain of a function from any representation.
• Relate the domain to context, explaining restrictions as a result of the context.
Related Standards: Current Course
Related Standards: Future Courses
I.A.CED.2, All functions standards (domain is used throughouthigh school mathematics courses)
II.A.CED.2, All functions standards (domain is used throughouthigh school mathematics courses)
Support for Teachers
Critical Background Knowledge
• Familiarity with function notation and domain (I.F.IF.2)
• Understandthedefinitionoffunction(8.F.1andI.F.IF.1)
• Independent, dependent variables and input/output (8.F.1)
Academic Vocabulary
Domain, function
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Interpret linear or exponential functions that arise in applications in terms of a context (F.IF.4-6)
Standard I.F.IF.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.
Concepts and Skills to Master
• Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Focus on linear and exponential functions.
• Estimate the rate of change from a graph.
Related Standards: Current Course
Related Standards: Future Courses
I.F.IF.9, I.F.LE.1, I.F.LE.3, I.S.ID.6, I.S.ID.7
II.F.IF.6, II.F.IF.9, II.F.LE.3, III.F.IF.6, III.F.IF.9, III.F.LE.3
Support for Teachers
Critical Background Knowledge
• Determine the rate of change from a description of a relationship or from two (x,y) values and interpret its meaning (8.F.4)
• Explain the slope m between any two points on a non-vertical line (8.EE.6)
Academic Vocabulary
Average rate of change, interval
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Interpret linear or exponential functions that arise in applications in terms of a context (F.IF.4-6)
Standard I.F.IF.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 functions and show intercepts.
e. Graph exponential functions, showing intercepts and end behavior.
Concepts and Skills to Master
• Given an equation of a linear or exponential function, create a graph by hand and show key features (intercepts, end behavior).
• Given an equation of a linear or exponential function, create a graph with technology and show key features (intercepts, end behavior).
Related Standards: Current Course
Related Standards: Future Courses
I.F.IF.4, I.F.IF.5, I.F.IF.6, I.A.REI.6, I.A.REI.11, I.A.REI.12, I.F.BF.3
II.F.IF.4, II.F.IF.7a, b, III.F.IF.4, III.F.IF.7b, c, d, e
Support for Teachers
Critical Background Knowledge
• Graph linear functions (8.EE.5, 8.F.3, and 8.F.5)
Academic Vocabulary
Linear, exponential, intercept, end behavior
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Analyze linear or exponential functions using different representations (F.IF.7,9)
Standard I.F.IF.9: Compare properties of two functions, each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Forexample, compare the growth of two linear
functions, or two exponential functions such as y=3n and y=100•2n.
Concepts and Skills to Master
• Compare properties of two functions, keeping the following in mind:
o properties include rate of change, intercepts, end behavior
o function pairs include linear to linear, linear to exponential, exponential to exponential
o representations include algebraically, graphically, numerically in tables, or by verbal descriptions
Related Standards: Current Course
Related Standards: Future Courses
I.F.IF.4, I.F.IF.7, I.F.LE.3, I.S.ID.7
II.F.IF.4, II.F.IF.7, II.F.LE.3, III.F.IF.4, III.F.IF.7, III.F.LE.3
Support for Teachers
Critical Background Knowledge (Activating prior knowledge)
• Compare properties of two functions (linear to linear), each represented in a different way (8.F.2)
• Interpret the equation of y = mx+b as defining a linear function (8.F.3)
• Construct a function, determine and interpret a rate of change and initial value of a linear function (8.F.4)
• Analyze graphs (increasing, decreasing, linear or nonlinear) (8.F.5)
Academic Vocabulary
function, slope, rate of change, intercept, interval, growth rate
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5630#71625
Building Linear or Exponential Functions Core Guide
Build a linear or exponential function that models a relationship between two quantities (F.BF.1-2)
Standard I.F.BF.1: 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. Forexample,builda functionthatmodelsthe temperature of a cooling body byadding a constant function to adecayingexponential, andrelatethese functions to themodel.
Concepts and Skills to Master
• Recognize patterns in a table, geometric model, or other representation to determine an explicit expression and a recursive process.
• Combine constant, linear, and/or exponential functions using addition or subtraction.
Related Standards: Current Course
Related Standards: Future Courses
All function standards, I.A.CED.2, I.A.SSE.1, I.G.CO.2,I.G.CO.4, I.G.CO.6
II.F.BF.1, II.F.BF.3, III.F.BF.1, III.F.BF.3, III.F.BF.4
Support for Teachers
Critical Background Knowledge (Access Background Knowledge)
• Generate and analyze patterns and relationships (4.OA.5, 5.OA.3)
• Simplifying expressions (7.EE.2)
• Apply integer exponent properties (8.EE.1)
• Definition of function (8.F.3)
• Construct linear functions (8.F.4)
• Describe relationships between quantities (8.F.5)
Academic Vocabulary
Function, intercepts, explicit expression, recursive
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70258
I.F.BF.1
Building Linear or Exponential Functions Core Guide
Build a linear or exponential function that models a relationship between two quantities (F.BF.1-2)
Standard I.F.BF.2: Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Limit to linear and exponential functions. Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.
Concepts and Skills to Master
• Write recursive and explicit formulas to represent arithmetic sequences.
• Write recursive and explicit formulas to represent geometric sequences.
• Connect arithmetic sequences to linear functions and geometric sequences to exponential functions.
• Translate between recursive and explicit formulas for sequences.
• Model contextual situations with arithmetic or geometric sequences.
Related Standards: Current Course
Related Standards: Future Courses
All function standards, I.A.CED.2
II.F.BF.1, III.A.SSE.4
Support for Teachers
Critical Background Knowledge
• Generate and analyze patterns and relationships (4.OA.5, 5.OA.3)
• Simplifying expressions (7.EE.2)
• Apply integer exponent properties (8.EE.1)
• Definition of function (8.F.3)
• Construct linear functions (8.F.4)
• Describe relationships between quantities (8.F.5)
Academic Vocabulary
Arithmetic sequence, geometric sequence, recursive, explicit
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70258
I.F.BF.2
Building Linear or Exponential Functions Core Guide
Build new functions from existing functions (F.BF.3)
Standard I.F.BF.3: Identify the effect on the graph of replacing f(x) by f(x) + k, for specific values of k (both positive and negative); find the value of k given the graphs. Relate the vertical translation of a linear function to its y-intercept. Experiment with cases and illustrate an explanation of the effects on the graph using technology.
Concepts and Skills to Master
• Describeverballyandgraphicallywhatwillhappentolinearandexponentialfunctionswhenf(x) isreplaced byf(x)+k,wherekisany realnumber.
• Given a graph of f(x) and f(x) + k on the same coordinate axis, find the value of k.
• Relate the vertical translation of a linear function to its y-intercept.
Related Standards: Current Course
Related Standards: Future Courses
I.A.SSE.1, I.F.IF.2, I.F.IF.4, I.F.IF.7, I.F.IF.9, I.F.LE.2, I.F.LE.5, I.G.CO.2, I.G.CO.4, I.G.CO.6
II.A.SSE.1, II.F.BF.3, II.F.IF.7, II.F.IF.8, III.A.SSE.1, III.F.BF.3, III.F.IF.7, III.F.IF.8
Support for Teachers
Critical Background Knowledge
• Graphing linear (8.F.3) and exponential functions(I.F.IF.7)
• Definition of function (8.F.3)
• Construct linear functions (8.F.4)
Academic Vocabulary
Transformation, Translation, vertical shift
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70258
I.F.BF.3
Linear and Exponential Core Guide
Construct and compare linear and exponential models and solve problems (F.LE.1-3)
Standard I.F.LE.1: Distinguish between situations that can be modeled with linear functions and withexponential functions.
a. Prove that linear functions grow by equal differences over equal intervals; 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
relativetoanother.
Concepts and Skills to Master
• Justify the fact that linear functions grow by equal difference over equal intervals using tables and graphs.
• Justify the fact that exponential functions grow or decay by equal factors over equal intervals using tables and graphs.
• Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
• Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative toanother.
Related Standards: Current Course
Related Standards: Future Courses
I.A.SSE.1, I.F.LE.2, I.F.LE.3, I.F.LE.5, I.F.IF.3, I.F.IF.6, I.F.BF.1, I.F.BF.2
II.A.SSE.1, II.F.IF.3, II.F.IF.4, II.F.IF.6, II.F.IF.9, II.F.BF.1, II.F.LE.3,
III.F.LE.3, III.F.LE.4, III.F.LE.5, III.A.SSE.1, III.F.IF.3, III.F.IF.4, III.F.IF.6, III.F.IF.9, III.F.BF.1, P.F.BF.1
Support for Teachers
Critical Background Knowledge
• Use proportional relationships to solve percent problems (7.RP .3)
• Describe where a function is increasing or decreasing (8.F.5)
• Identify the constant rate of change (7.RP.2b, 8.EE.5, 8.F.4, 8.F.5)
• Find a percent of a quantity as a rate per 100 (6.RP .3c)
Academic Vocabulary
interval, rate, factors, constant rate of change, percent rate per unit, growth, decay
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70276
I.F.LE.1
Linear and Exponential Core Guide
Construct and compare linear and exponential models and solve problems (F.LE.1-3)
StandardI.F.LE.2:Constructlinearandexponentialfunctions,includingarithmeticand geometricsequences, givenagraph,adescription ofarelationship,ortwoinput-output pairs(includereadingthesefromatable).
Concepts and Skills to Master
• Construct a linear function and/or an arithmetic sequence given a situation, a set of ordered pairs, or a table.
• Construct an exponential function and/or a geometric sequence given a situation, ordered pairs, or a table.
Related Standards: Current Course
Related Standards: Future Courses
I.A.SSE.1, I.F.LE.1, I.F.LE.3, I.F.LE.5, I.F.IF.2, I.F.IF.3, I.F.IF.6, I.F.BF.1, I.F.BF.2
II.A.SSE.1, II.F.IF.3, II.F.IF.4, II.F.IF.6, II.F.IF.9, II.F.BF.1, II.F.LE.3,
III.F.LE.3, III.F.LE.4, III.F.LE.5, III.A.SSE.1, III.F.IF.3, III.F.IF.4, III.F.IF.6, III.F.IF.9, III.F.BF.1, P.BF.1
Support for Teachers
Critical Background Knowledge
• Construct a function to model linear situation (8.F.4)
• Use function notation (I.F.IF.2)
Academic Vocabulary
Exponential, linear, arithmetic, geometric, sequences
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70276
I.F.LE.2
Linear and Exponential Core Guide
Construct and compare linear and exponential models and solve problems (F.LE.1-3)
StandardI.F.LE.3:Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceeds aquantityincreasing linearly.
Concepts and Skills to Master
• Observe that a quantity increasing exponentially eventually exceeds a quantity increasing linearly using graphs and tables.
Related Standards: Current Course
Related Standards: Future Courses
I.A.REI.6, I.F.IF.6, I.F.IF.7, I.F.IF.9, I.F.LE.1, I.F.LE.2, I.F.LE.5
II.A.REI.7, II.F.IF.4, II.F.IF.6, II.F.IF.7, II.F.IF.9, II.F.LE.3, II.F.IF.6, III.F.LE.3, P.F.IF.7
Support for Teachers
Critical Background Knowledge
• Perform operations using whole number exponents (6.EE.2c)
• Identify, compare, and interpret rates of change (7.RP.2b, 8.F.2, 8.EE.5)
• Identify linear and nonlinear functions from a graph or a table (8.F.4, 8.F.5)
Academic Vocabulary
Linear, exponential, increasing
Resources
Curriculum Resources: http://www.uen.org/core/core.do?courseNum=5600#70276
I.F.LE.3
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