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Unit Topics These modeling standards need to be embedded through the course: Code *N.Q.3 Use units as a way to understand problems and to guide the solution of multi-step problems;choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. Define appropriate quantities for the purpose of descriptive modeling. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections or complements of other events (or, and, not). N.RN.3 Use properties* of rational and irrational numbers. Explain why the sum or product of rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational. (closure, commutative, distributive, identity, inverse) *N.Q.1 *N.Q.2 1 CCLS Real Numbers 8 Days 1. Review Prior Knowledge* ( 4 days) a. Number Sets* b. Operations with Real Numbers* c. Order of Operations* d. Evaluating Algebraic Expressions* 2 . Set Theory ( 2 days) a. Review subsets of Real Numbers* b. Use set builder and/or interval notation ( Ex: {x | x > 4}, {x | 4 < x < 10}, {x | x < -3 or x > 11}) c. Define and solve graphic, letters and numbers in problems using the following: 1. roster notation 2. subset 3. universal set 4. complement 5. null set 6. intersection 7. union 3 . Properties of Real Numbers (1 day) a. Commutative* b. Associative* c. Identity* d. Inverse* e. Distributive* f. Zero Property* g. Closure 4. Review and Assessment (1 day) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 2 Polynomials 13 days 1. Operations With Monomials and Polynomials (9 days) a . Add and subtract monomials* b . Add and subtract polynomials* c . Multiply and divide monomials *(include negative exponents) A.APR.1 N.RN.2 Understanding that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, multiplication; add, subtract and multiply polynomials. Rewrite expressions involving radicals and rational exponents using the properties of exponents. d . Distributive Property (include combining like terms) e . Multiply two binomials* f . Multiply two polynomials g . Multiply and divide monomial expressions with a common base using the properties of exponents (use integers as exponents)* Interpret expressions that represent a quantity in terms of its context. Interpret parts of an expression such as terms, factors, and A.SSE.1a coefficients. A.SSE.1 Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as A.SSE.1b a product of p and a factor not depending on P. h . Divide a polynomial by a monomial or binomial, where the quotient has no remainder* [**New] [**8th] [**8th] [**8th] [**8th] [**8th] i. Divide simple rational expressions where the quotient does have a remainder A.APR.6 Rewrite simple rational expressions in different forms; Write a(x)/b(x) in the form q(x) + r(x)/b(x) where a(x), b(x), q(x), r(x) are polynomials with the degree of r(x) is less than the degree of b(x) using inspection, long division, or, for the more complicated examples, a computer algebra system. [?] 2. Scientific Notation (2 days) a. Convert to scientific notation b. Convert from scientific notation to standard form c. Compute products and quotients involving scientific notation d. Solve word problems involving products and quotients 3 . Review and Assessment (2 days) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 3 Linear Equations A.CED.1 12 days A.REI.3 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions, and simple rational and exponential functions. Solving linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Contruct a viable argument to justify a solution method. A.CED.4 Rearrange formulas to highlight a quatity of interest, using the same reasoning as in solving equations. 1. Translate Among Verbal, Written, and Algebraic (2 days) a. Represent words with mathematical expressions and equations (and vice versa)* b. Distinguish between an algebraic expression and an algebraic equation c. Write algebraic equations to represent a situation 2 . Solve Linear Equations (2 days) a. Solve one-and two-step equations with one unknown (include fractions and decimals) 1. Combine like terms* 2. Use Distributive Property* 3. Variables on opposite sides of the equal sign* 4. Fractional equations 5. Variable in the denominator 6. Equations with an infinite number of solutions such as 2x+4 = 2(x+2) 7. Equations with no solutions such as x + 4 = x + 5 8. Have students construct a viable argument to justify a solution method. b. Solve an algebraic proportion with one unknown that results in a linear equation* c. Determine whether a value is a solution to a given linear equation d. Solve literal equations for a given variable (Ex: pq + r = n, solve for q). Algebra / Word Problems with Fractions, Decimals, Percents and Proportions 3. (4 days) a. Analyze and solve word problems whose solution requires solving a linear equation with one unknown (consecutive integer, perimeter and so on. b. Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation*. 4. Calculate Rates Using Appropriate Units (1 day) a. Determine appropriate units* b. Compare rates using proportions* [**8th] [**8th] [**8th] 5. Conversions Within Measurement Systems (1 day) a. Convert between units* b. Solve measurement problems given more than one unit* 6. Review and Assessment (2 days) Common Assessment # 1 ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 4 Linear Inequalities A.REI.3 Solving linear equations and inequalities in one variable, including equations with coefficients represented by letters. 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 different foods. 4 days 1. Solve Linear Inequalities (3 days) a. Solve linear inequalities with one variable (include multiplying and dividing by a negative number)* b. Translate among verbal, written and algebraic* 1. Represent words with mathematical inequalities 2. Write algebraic inequalities that represent a situation c. Determine whether a given value is a solution to a linear inequality with one variable d. Analyze and solve word problems whose solution requires solving a linear inequality with one variable 2. Review and Assessment (1 day) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 5 Graphs of Linear Equations and Inequalities 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 different foods. 14 days 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). 1. Graphs of Lines (10 days) a. Graph a line using a table of values* F.IF.1 b. Define a function* and function notation F.IF.2 1 .Ordered pairs, domain, range 2. Vertical line test c. Determine whether a given point is on a line, given an equation. ( Ex: Given (3, 2) is on the graph of x + 3y = k, find k ) F.IF.5* Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 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) give the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. d. Solve for an unknown in an equation (or for a coordinate) given a point on a graph ( Ex: Given (k, 7) is on the graph of x – 7y = 10, find k ) e. Slope* 8.EE.6 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. Use similar triangles to explain why the slope, m, is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y=mx for a line through the origin and the equation y=mx+b for a line intercepting vertical axis at b. [?] F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. S.ID.7 F.LE.1b 1. Define slope as the average rate of change* presented symbolically or as a table 2. Find slope between two points using the graph.* 3. Find slope between two points using the slope formula. 4. Identify different slopes (positive, negative, zero and undefined) f. Intercepts (x and y) 1. Find the x- and y-intercepts (*y-intercept only) 2. Write an equation and graph lines parallel to the axes 3. Graph a line using the x and y intercepts (optional) g. Slope-Intercept Form 1. Write the equation of a line given its slope and y-intercept* 2. Convert an equation into slope-intercept form 3. Graph equations using slope-intercept form* (include investigating and generalizing how changing the slope and y-intercept affects the graph) 4. Determine if two lines are parallel, given their equations in any form 5. Write the equation of a line, given the coordinates of two points on the line h. Write the equation of a line given the slope and the coordinates of a point. (point-slope form) 2. Graphs of Linear Inequalities (2 days) a. Determine if reference line is dashed or solid b. Determine where to shade 3. Read and interpret graphs. 4. Review and Assessment (2 days) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 6 Systems of Linear Equations and Inequalities 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 different foods. 10 days 1. Systems of Equations (6 days) A.CED.2 A.REI.6 a . Solve algebraically and check 1. substitution method 2. elimination method b. Solve graphically and check* c. Solve word problems 2. Systems of Inequalities (2 days) a. Solve graphically b. Determine whether a given point is in the solution set 3. Review A.REI.5 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Solve exactly systems of linear equations and approximately (eg., with graphs), focusing on pairs of linear equations in two variables. 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 solution. [?] 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 A.REI.12 corresponding half-planes. and Assessment (2 days) Common Assessment # 2 ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. Order of Chapters 7 and 8 left to the discretion of the teacher. 7 Factoring and Algebraic Fractions A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represent by the expression. A.SSE.2 Use the structure of an exxpression to identify ways to rewrite it. For example, x 4 -y 4 as (x 2 ) 2 -(y 2 ) 2 thus recognizing it as the difference of two squares that can be factored as (x 2- y 2 )(x 2 +y 2 ). 13 days 1. Factoring (4 days) a. Factor algebraic expressions using the GCF* b. Factor a trinomial into two binomials, given a = 1* c . Factor a trinomial into two binomials (a >1 ??) d. Identify and factor the difference of two perfect squares e. Factor a trinomial into binomials that requires factoring out a GCF first f. Factor a trinomial into two binomials that requires factoring out a negative GCF first 2. Algebraic Fractions (5 days) a. Find the value for which an algebraic fraction is undefined monomial and binomial denominators b. factorable binomial denominators 1. Simplify fractions with polynomials in the numerator and denominator by factoring both and reducing to lowest terms c. Add or subtract fractional expressions 1. like and unlike monomial denominators 2. like binomial denominators d. Multiply and divide algebraic fractions and express the product or quotient in simplest form 1. monomial divided by monomial* 2. polynomial divided by monomial* 3. polynomial divided by polynomial 3. Rational equations and radical equations in one variable(2 days) A.REI.2 A.CED.1 4. ROUGH DRAFT Solve simple rational and radical equations in one variable, and give examples showing how extraneous soltions arise. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions, and simple rational and exponential functions. Review and Assessment (2 days) LAST PRINTED ON 2/9/2012 AT 3:59 PM. 8 Non-linear Functions F.IF.1 23 days F.IF.2 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y=f(x). Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. A.REI.4 Solve quadratic equations in one variable. 1. Graph Quadratic Equations (2 days) a. Recognize general form of a quadratic* b. Determine the axis of symmetry using the formula or from the graph c. Determine the vertex algebraically or graphically d. Find the table of values for an equation e. Graph the equation 2. Solve Quadratic Equations (7 days) a. Identify the relationship between the roots of a quadratic equation and the xintercepts on the graph of the parabola [**New] [**New] A.SSE.3a b. Find the roots of a parabolic function graphically (integer solutions) c. Find the roots of a parabolic function algebraically A.REI.4b 1. Taking square roots 2. Completing the square 3. Quadratic formula d. Identify the relationship between the number of roots and the number of xintercepts A.REI.4b e. Understand and apply the multiplication property of zero to solve quadratic equations whose coefficients and roots are integers f. Analyze and solve word problems that involve quadratic equations Factor a quadratic expression to reveal the zeros of the function it defines. Solve quadratic equations by inspection (eg., for x 2 =49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. [?] Solve quadratic equations by inspection (eg., for x^2=49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. F.IF.5* Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. 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) give the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. A.REI.7 Solve a simple system consisting of a linear equaiton and a quadratic equation in two variables algebraically and graphically. F.IF.7b Graph square root, cube root functions, including step functions and absolute value functions. A.APR.3 g. Solve algebraic proportions with one unknown that result in quadratic equations 3. Solve Systems of Linear and Quadratic Equations Algebraically ( 1 day) a. Solve algebraically using substitution 4. Families of Functions (10 days) a. Identify and graph all types 1. List of functions - exponential - absolute value - polynomial (linear, quadratic, cubic) - square root - cube root - step 2. Analyze/ Compare/ Interpret functions graphically, algebraically, verbally and numerically in tables. - average rate of change (over specified interval) - domain and range - roots - end behavior b. solve systems of functions graphically for points of intersection c. Exponential growth/decay F.IF.6 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal desriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. 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. F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. F.IF.9 5. Review and Assessment ( 3 days) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 9 Radicals and Right Triangles 15 days N.RN.1 [**New] 1. Operations with Radicals (5 days) a. Simplify radicals 1. perfect square radicand* 2. radicand with a perfect square factor (no variable) b. Add and subtract like and unlike radicands c. Multiply radicals (include use of distributive property) d. Divide radicals e. Fractional exponents Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example,we define 5 (1/3) as the cube root of 5 since (5 1/3 ) 3 is 5. [**New] 2. Solve simple radical equations where the radicand is a linear binomial A.REI.2 [?] Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions arise. 3. Pythagorean Theorem (2 days) a. Determine if a triangle is a right triangle* b. Find the third side (radical, rational and irrational) of a right triangle, given two sides* c. Solve word problems *G.SRT.8 Using trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. *G.SRT.8 Using trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. 4. Trigonometric Functions (5 days) a. Using similar triangles to introduce the three trig ratios: 1. Sine and its Inverse 2. Cosine and its inverse 3. Tangent and its inverse b. Find all sides and angles of a right triangle using trig. and/or Pythagorean Theorem * c. Solve word problems (include angle of elevation and depression) * 5. Review and Assessment (3 days) Common Assessment # 3 ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. 10 Plane and Solid Geometry Assume Perimeter, Area, Volume of rectangular shape used in word problems. 12 days 1. Perimeter (2 days) [**6th] [**6th] [**6th] [**6th] [**6th] [**6th] [**7th] [**7th] [**7th] NOTE: Include problems that use algebra to find / express perimeter of the following figures: a. Triangle b. Rectangle c. Square d. Parallelogram e. Rhombus f. Trapezoid g. Circle h. Semi-Circle i. Quarter- Circle j. Other Regular Polygons k. Figures composed of polygons and/or circles or sectors of a circle 2. Area (4 days) [**6th] [**6th] [**6th] [**6th] [**6th] [**6th] [**7th] [**7th] [**7th] [**??] [**??] NOTE: Include problems that use algebra to find /express area of the following figures: a. Triangle b. Rectangle c. Square d. Parallelogram e. Rhombus f. Trapezoid g. Circle h. Semi-Circle i. Quarter-Circle j. Figures composed of polygons and/or circles or sectors of a circle k. Shaded regions 3. Volume (2 days) [**6th] [**6th] [**8th] [**6th] [**6th] [**6th] [**6th] [**??] a. Cube b. Rectangular Prism c. Cylinder-Cone-Sphere 4. Surface Area (2 days) a. Cube b. Rectangular Prism c. Cylinder 5. Relative Error (1 day) a. Calculate the relative error in measuring square and cubic units, where there is an error in the linear measure 6. Review and Assessment (1 day) ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM. [?Permutations and Factorials?] 11 Probability 7 days 1. Probability (7 days) a. Define and solve problems using the following 1. sample space 2. simple probability of a single event 3. probability with “and” (single event) 4. probability with “or” (single event) 5. complement S.CP.5 Recognize and explain the concepts of conditional probability and independence in everyday languate and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. S.CP.1 Describe events as subsets of a sample space using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of the other events. S.CP.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the twoway table as a sample space to decide if events are independent and to approximte conditional probabilities. 6. empirical probability (based on specific sample data)* 7. impossible events 8. certain events 9. Fundamental Counting Principle b. Analyze a set of events to determine when: 1. some or all are equally likely to occur 2. one is more likely to occur 3. an event is certain to happen or not to happen S.CP.6 Understand the conditional probability of A given B as P(A and B)/P(B) and interpret independence of A and B as saying the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. c. Conditional Probability S.CP.3 1. with replacement 2. without replacement d. Probability of a series of independent and dependent events 1. and 2 . or 3. mutually exclusive 4. not mutually exclusive [**??] [**??] [**??] [**??] [**??] [**??] [**??] 2. Permutations (2 days) a. Review definition of factorial b. Formula 3. Factorials (2 days) a. Define factorial b.. Evaluate factorials c. Simplify expressions with factorials (rational, algebraic) 4. Review ROUGH DRAFT and Assessment (2 days) LAST PRINTED ON 2/9/2012 AT 3:59 PM. 12 Statistics and Regression S.ID.1 11 days S.IC.3 S.ID.3 Interpret differences in shapes, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). S.IC.1 1. Categorize Data (1 day) a. Qualitative b. Quantitative c. Univariate d. Bivariate Represent data with plots on the real number line (dot plots, histograms, and box plots) Understand statistics as a process for making inferences about populations parameters based on a random sample from that population. Recognize the purposes of and differences amoung sample surveys, experiments, and observational studies; explain how randomization relates to each. 2. Measures of Central Tendency (1 day) a. Mean b. Median c. Mode d. Appropriate choice of measure* e. Range* f. Recognize how linear transformations of one-variable data affect the data’s mean, median, mode and range 3. Histograms (2 days) a. Generate a frequency table b. Construct a frequency histogram c. Construct a cumulative frequency histogram d. Analyze data from a frequency table, histogram, cumulative frequency table or cumulative frequency histogram 4. Percentiles (1 day) a. Definition b. Find the percentile rank of a value in a set of a data c. Find the first, second and third quartiles 5. Box-and-whisker Plot (1 day) a. Define as a five-number summary (minimum, maximum, quartiles) b. Construct a box-and-whisker plot c. Analyze data from a box-and-whisker plot 6. Biased Data (1 day) a. Evaluate reports and graphs for accuracy, appropriateness, experimental design and soundness of the conclusion b. Identify and describe sources of bias and its effect, drawing conclusions from data a. Construct a scatterplot S.ID.6 b. Identify the independent and dependent variables c. Find the relationship between the variables (positive, negative, none) S.ID.6a S.ID.6c d. Understand the difference between correlation and causation e. Identify variables that might have a correlation but not a causal relationship f. Construct manually a line of best fit S.ID.7 S.ID.9 Interpret differences in shapes, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. [?] Fit a linear function for a acatter plot that suggests a linear association. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Distinguish between correlation and causation. S.ID.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit. 7. Scatterplots (2 days) [**New] g. Determine the equation of a line of best fit h. Interpolation and extrapolation i. Correlation coefficient S.ID3 8. Review and Assessment (2 days) 13 Regents Review 15 days ROUGH DRAFT LAST PRINTED ON 2/9/2012 AT 3:59 PM.