Number and Quantity and Review



Algebra 1 StandardsNumber and Quantity and ReviewI can...add, subtract, multiply, and divide fractionsThe goal is for students to develop proficiency with operations on fractions, without using a calculator with fraction capabilities. Students may use a calculator to find a common multiple or pare and order real numbers with and without a number line (A1.2.A)choose and interpret units consistently in formulas, and the origin in graphs and data displays (N.Q.1)The goal is for students to be aware of how they are using and interpreting formulas, and to know what their answer means contextually. This standard should be embedded within all tasks throughout the year.define appropriate quantities for the purposes of descriptive modeling and choose a level of accuracy appropriate to limitations on measurement (N.Q.2–3; A1.2.D)The goal is for students to understand the limitations inherent in their modeling choices. Students should understand that all measurements include a degree of error and that the tools they use effect the precision they can obtain. Students should understand the difference between exact results and approximations and make conscious decisions about when to use each.This standard should be embedded within all tasks throughout the year.Exponents and Square RootsI can...use the properties of exponents to rewrite, simplify, and evaluate exponential expressions (N.RN.2; A1.2.C)Students should know the rules for multiplying and dividing two exponential expressions with common bases, and for raising a power to a power.For N.RN.2, students should be comfortable moving between an expression with rational exponents and an equivalent expression with radicals.use properties of square and cube roots to evaluate or rewrite in equivalent and simplified forms (N.RN.2; A1.2.C)Students should be comfortable simplifying square and cube roots and performing operations, including rationalizing the denominator, with square root expressions.Students should know that the nth root of xm can be expressed as xm/n.Linear EquationsI can...create linear equations to represent relationships between quantities (A.CED.1–3; F.LE.1–2; A1.1.A–B; A1.4.A–B)Students should be able to write the equation represented by a word problem or other context. In-class work should emphasize interpreting and using these equations within the given context.Students should be able to write equations given: graphs, verbal descriptions of relationships, two input-output pairs, a table with input-output pairs.Students should be able to use point-slope form to find the equation of a line.Students should be able to find the slope of a line parallel or perpendicular to a given line.Students should be comfortable moving between standard form, point-slope form, and slope-intercept form.solve linear equations (A.CED.1, 4; A.REI.1, 3; A1.4.A; A1.7.D)Students should be able to solve multi-step equations, including those that require both the distributive property and combining like terms.Students should be able to rewrite an identity or formula that is linear in the variable of interest.For example: solve V=IR for R; solve e = mc2 for m; solve y-y1=m(x - x1) for m; solve 2x+3y=12 for y.determine the slope between two points on a given line (F.IF.6; A1.4.C)Students should be able to do this algebraically, and to determine the points from a table or context.find the x- and y-intercepts of a linear equation (F.IF.7a; A1.4.C)Students should be able to do this graphically, and algebraically.graph linear functions (F.IF.7a; A1.4.B)Students should be able to do this given a table of input-output pairs, from a context with input-output pairs embedded within it, and from an equation (in any form).interpret x- and y-intercepts and slope of a linear function in context (F.IF.4; F.LE.5; A1.4.C, E)For slope, the emphasis is on the rate of change between the dependent and independent variables.For intercepts, the emphasis is on understanding the relationship between the values of the dependent and independent variables; i.e., when the independent variable takes on a special value, it tells us something about the dependent variable’s value at that time.Exponential EquationsI can...create exponential equations to represent relationships between quantities (A.CED.1–2; F.LE.1–2)Students should be able to write simple exponential equations of the form y=bn given a table of values, graph, or contextual situation. This standard will be further explored during Algebra II.For example: A certain population of bacteria doubles every minute. Write an equation that represents the number of bacteria at minute t if the population at t=0 is 25.solve simple exponential equations (A.CED.1–2, 4; A.REI.1; A1.1.E; A1.7.B)Students should be able to solve for n or y in an equation of the form y = abn, where n is an integer, graphically and/or numerically as appropriate. When possible, students should find exact answers, otherwise students should be able to find approximate answers.interpret the parameters of an exponential function in terms of a context (F.IF.4; F.LE.5; A1.7.A)Students should be able to identify the effect that a and b have on the graph of y=abn, when n is an integer.sketch the graph of an exponential function (F.IF.7e; A1.7.A)Students should be able to graph an exponential of the form y=abn where n is an integer.Quadratic EquationsI can...create quadratic equations to represent relationships between quantities (A.CED.1–2; A1.5.A)Students should be able to represent quadratic functions symbolically, graphically, verbally, and in a table. In-class work should emphasize interpreting and using these equations within the given context.For example: given a quadratic function representing the height of a thrown object over time, make a table, sketch a graph, give a verbal description of the graph, and determine the height of the object after a given number of seconds.For example: given the area of a rectangle and two expressions in x for the length and width, write and solve a quadratic equation to find the length and width.solve quadratic equations by taking square roots (A.CED.4; A.REI.4b; A1.1.D)Solutions should be real numbers. Complex roots will be explored in Algebra II.For example: If x2=25, what are appropriate values for x?solve quadratic equations by completing the square (A.CED.4; A.REI.4a; A1.1.D; A1.5.D)Solutions should be real numbers. Complex roots will be explored in Algebra II.solve quadratic equations using the quadratic formula (A.REI.4b; A1.1.D; A1.5.D)Solutions should be real numbers. Complex roots will be explored in Algebra II.Students should derive the quadratic formula by completing the square.solve quadratic equations by factoring (A.CED.4; A.REI.4b; A.SSE.2; A1.1.D; A1.5.C)Solutions should be real numbers. Complex roots will be explored in algebra II.Equations should be factorable as (ax+b)(cx+d), where a, b, c, and d are integers.symbolically and graphically identify zeroes, extreme values, and symmetry of a quadratic equation and interpret these in terms of a context (F.IF.4, 7a, 8a; A1.5.B)Students should be able to identify the effects of the parameters in quadratic equations of the form: ax2+c, ax-h2+k, and a(x-r)(y-s).Weird FunctionsI can...graph and identify key features of square root and cube root functions (F.IF.7b)Students should be able to graph a square root or cube root function using technology or by making a table of values.Students should be able to identify domain and range and behavior as x goes to ±∞. Students should compare and contrast with linear, exponential, and quadratic functions.graph and identify key features of piece-wise defined functions (F.IF.7b)Students should be able to graph by hand a piece-wise defined function described verbally or algebraically by making a table of values.Students should be able to identify domain and range, and any other interesting features of the graph. Students should compare and contrast with linear, exponential, and quadratic functions.graph and identify key features of absolute value and step functions (F.IF.7b)Students should be able to graph an absolute value or step function by making a table of values.Students should be able to identify domain and range. Students should compare and contrast with linear, exponential, and quadratic functions.graph and identify key features of exponential and logarithmic functions (F.IF.7e)Students should be able to graph an exponential or logarithmic function using technology or by making a table of values.Students should be able to identify domain and range and behavior as x goes to ±∞. Students should compare and contrast with linear, exponential, and quadratic functions.graph and identify key features of trigonometric functions (F.IF.7e)Students should be able to graph a trigonometric function using technology.Students should be able to identify domain, range, period, midline, and amplitude. Students should compare and contrast with linear, exponential, and quadratic functions.FunctionsI can...determine if a given relation is a function (F.IF.1; A1.3.A)Students should understand that not all relations are functions, and that a function requires all elements of the domain to be mapped to exactly one element of the range.Relations should be given algebraically, graphically, verbally, in a mapping diagram, as ordered pairs, or in a table.determine the domain and range of a function and relate the domain of a function to its graph and the context it describes (F.IF.1, 5; A1.3.A)Students should be able to identify the domain and range of a function given algebraically, graphically, verbally, in a mapping diagram, as ordered pairs, or in a table.Students should be able to restrict the domain of a given relation so that it is a function. Students should be able to restrict the domain of a given function so that it makes sense for a given contextual situation.use function notation to evaluate functions for inputs in their domains and interpret statements that use function notation in a context (F.IF.2; A1.2.B; A1.3.C)Students should also be able to evaluate algebraic expressions for given values.identify and interpret key features of a function (F.IF.4; A1.3.A–B)Students should be comfortable moving between various representations of functions, including: symbolically, graphically, in a table, or in wordsKey features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums or minimums; symmetries; end behavior; and periodicity.Students should be introduced to a wide variety of functions, but an in-depth study is reserved for: linear, quadratic, and simple exponential equations. Other classes of functions will be developed more rigorously in Algebra II.calculate and interpret the average rate of change of a function over a certain interval (F.IF.6)Standard 9 (slope) develops this concept for linear equations.Functions may be given symbolically, graphically, in a table, or in words.The goal is for students to interpret the rate of change of various classes functions over pare properties of functions, including those represented in different ways (F.IF.9; A1.3.B; A1.4.E)The emphasis for this standard is on linear, exponential, and quadratic functions. Other classes of functions will be explored in more depth in Algebra II.Students should be able to compare a given linear or exponential function to its parent function.For example: Given the graph of a quadratic function and the equation of another, identify which has the larger maximum. Compare the growth between the functions y=3x and y=100x.write a function using an explicit expression, a recursive process, or steps for calculation from a context (F.BF.1)The goal is for students to be able to solve problems using functions of their own creation. These functions may be in the form of equations, but may also be algorithms or processes that lead to a correct solution. This standard should be embedded within all tasks throughout the year.identify and graph transformations of a function (F.BF.3)The emphasis for this standard is on linear, exponential, and quadratic functions. Other classes of functions will be explored in more depth in Algebra II.Students should be able to identify vertical and horizontal shifts and dilations, as well as reflections about either axis.find the inverse of a given function (F.BF.4a)Students should be able to find the inverse of a linear function. Other classes of functions will be explored further in Algebra II.Students should also be able to state what domain restrictions are necessary in order for the inverse to exist for a given quadratic function.interpret expressions, and parts of expressions, that represent a quantity in terms of its context (A.SSE.1)The goal is for students to think about more complicated expressions in terms of their component pieces. This standard should be embedded within all tasks throughout the year.choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression (A.SSE.3; F.IF.8)The goal is for students to think about the various ways expressions can be written, and what each reveals about the underlying context. This standard should be embedded within all tasks throughout the year, but goes particularly well with the unit on quadratic functions and factoring.Systems of EquationsI can...solve systems of two equations or inequalities algebraically and graphically (A.REI.5–7, 11; A1.1.C; A1.4.D)Students should be able to solve systems of two linear equations and systems of one linear and one quadratic equation.Students should be able to use elimination and substitution to solve systems of equations algebraically.Students should be able to identify when the two equations describe the same line or parallel lines, and what this means in terms of the number of solutions that exist.An emphasis should be placed on students understanding why their solutions methods are valid. Students should understand that, given a system of equations, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Students should understand 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 fx=g(x).InequalitiesI can...create inequalities in one variable and use them to solve problems (A.CED.1; A1.1.B; A1.4.A)In-class work should emphasize interpreting and using these equations within a given context.solve linear inequalities in one variable (A.REI.3; A1.4.A)Students should be able to solve compound and absolute value inequalities.Stuents should be able to solve linear inequalities with variable coefficients.graph the solutions to a linear inequality or set of linear inequalities in two variables (A.REI.12; A1.1.B–C)Students should know that a linear inequality in two variables describes a half-plane.represent constraints by inequalities, or systems of inequalities, and interpret solutions as viable or non-viable options in a modeling context (A.CED.3; A1.1.B)The emphasis for this standard is on linear inequalities. This standard will be explored in more depth in Algebra II.StatisticsI can...create, interpret, analyze, and use dot plots (S.ID.1)create, interpret, analyze, and use histograms (S.ID.1)create, interpret, analyze, and use box-and-whisker plots (S.ID.1)create, interpret, analyze, and use two-way frequency tables (S.ID.5)Students should be able to construct two-way frequency tables. An emphasis should be placed on interpreting the data and making decisions based on it.Students should be able to interpret relative frequencies in the context of the data and recognize possible trends and associations in the data.create, interpret, analyze, and use scatter plots (S.ID.6–8; A1.6.D–E)Students should be able to fit a function to the data and write an equation for that function. For S.ID.6, the function may be linear, quadratic, or exponential. For A1.6.D, the function should be linear.Students should understand how the slope and y-intercept of the line of best fit relates to the data.Students should compute (using technology) and interpret the correlation coefficient of a linear fitStudents should be able to assess the fit of a line of best fit by analyzing residuals.Students should be able to describe correlation in terms of strong or weak and positive or negative.An emphasis should be placed on using the line of best fit to make predictions.find and interpret summary statistics for given data sets (S.ID.2–3; A1.6.A, C)The goal is for students to analyze statistical data and make judgments or decisions on this basis. The standard should be embedded within tasks throughout the unit.Students should be able to describe how a linear transformation of the data would affect these summary statistics.Students should be able to find and interpret: mean, median, mode, range, interquartile range, outliers, and standard deviation.Students should be able to interpret differences in shape, center, and spread of data sets.make valid inferences and draw conclusions based on data (S.ID.9; A1.6.B)The goal is for students to use statistical measures, models, and displays to make decisions, predictions, and judgments. This standard should be embedded within tasks throughout the unit.Students should know the difference between correlation and causation, and identify invalid arguments that mistake one for the other.Sequences and SeriesI can...construct and use arithmetic sequences (F.BF.1a, 2; F.LE.2; A1.7.C)Students should understand that arithmetic sequences can be modeled linearly.Students should be able to write explicit and recursive forms given a graph, a description of a relationship, or two input-output pairs.Students should be able to use the explicit or recursive form of a sequence to identify a specific term in the sequenceconstruct and use geometric sequences (F.BF.1a, 2; F.LE.2; A1.7.C)Students should understand that geometric sequences can be modeled exponentially.Students should be able to write explicit and recursive forms given a graph, a description of a relationship, or two input-output pairs.Students should be able to use the explicit or recursive form of a sequence to identify a specific term in the sequencePolynomial ManipulationI can...add, subtract, and multiply polynomials (A.APR.1; A1.2.F)The emphasis for this standard is on polynomials of degree 2. This includes polynomials that simplify to degree 2. Polynomials of higher degree will be covered in Algebra II. Division of polynomials will be covered in Algebra bine like terms in a polynomial (A.SSE.3; A1.2.E)Students should know that like terms have the same base raised to the same power.Students should be able to combine terms raised to arbitrarily large powers.factor a polynomial (A.SSE.3; A1.2.E)Possible methods of factoring include: factoring a monomial from a polynomial; difference of squares; perfect square trinomials; quadratic trinomials; trinomials that are the product of a constant and a trinomial.For this course, an emphasis should be placed on factoring quadratic expressions, including those which require completing the square.Emphasis should be placed on both computational fluency and conceptual understanding.Miscellaneous Mathematical KnowledgeI can...determine all possible values of variables that satisfy prescribed or contextual conditions (A1.2.B)The goal is for students to think about the context of the equations and expressions they use. This standard represents both mathematical constraints (in the expression 1/x, x cannot be 0) and contextual constraints (within a problem, what values do and do not make sense).This standard should be embedded within all tasks throughout the year.For example: If the function d(t)=.02t represents the distance in miles from the starting line of a runner after t minutes, what are all of the valid values for t? ................
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