矍 Smithtown Central School District 8 Honors Integrated Algebra Curriculum August 2007 I. Review of Real Number System (7 Days) II. III. IV. Algebraic Expressions and Linear Equations (10 Days) Inequalities (4 Days) Exponents and Monomials (9 Days) V. Polynomials (6 Days) VI. VII. VIII. IX. Factoring (7 Days) Solving Quadratic Equations by Factoring (6 Days) Ratio, Proportion and Percent (11 Days) Algebraic Fractions/Rational Expressions (8 Days) X. Plane and Solid Geometry (7 Days) XI. XII. XIII. XIV. XV. XVI. XVII. Transformational Geometry (Take Home Packet) Functions (10 Days) Radicals (7 Days) Right Triangle Trigonometry (5 Days) Graphing Linear Equations (7 Days) Writing Linear Equations (6 Days) Systems of Equations and Inequalities (8 Days) 诲꿬矑矓 矓 Ɉ XVIII. Graphing Quadratic Equations (8 Days) XIX. XX. Probability (9 Days) Statistics (10 Days) 1
Math 8 Honors Curriculum (Updated August 2007) I. Review of Real Number System (8 days) Technology: graphing calculator Number Sets (real, whole, integers, rationals, irrationals) Review of integer & rational number operations Properties (commutative, associative, distributive, identities, inverses and closure) A.N.1 Order of operations (including integers, decimals, fractions) Evaluating expressions (including absolute value & exponents) A.N.6 Set Theory (no more than three sets) a. Elements of a Set A.A.29 Set Builder Notation Example Set Builder Notation: {x x is a counting number} Interval Notation Example Interval Notation: [ 3,12) ( -3 is included 12 is not ) b. Kinds of Sets Finite, Infinite, Empty, Null c. Operations with Sets Complement of a given subset A.A.30 Intersection of sets A.A.31 Union of sets A.A.31 Binary Operations and properties of real numbers Absolute Value Additive Inverse Associative Closure Binary Operations Commutative Complement Distributive Element Empty Set Equation Evaluate Expression Finite Sample Space Group Identities Infinite Sample Space Infinite Set Integers Intersection of two Sets Interval Notation Irrational Multiplicative Inverse Null Set Rational Real Numbers Reciprocals Set Builder Notation Solve Subset Symmetric Union Universal Set Universe Whole Numbers Zero Property A.N.1 Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas. A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form A.A.30 Find the complement of a subset of a given set, within a given universe A.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets) Integrated Mathematics Course 1 Textbook; Chapters 1, 7, 19, 20-5 Algebra 1 Concepts and Skills Textbook; Chapters 2, 9, 12.6 http://www.math.com/school/subject3/lessons/s3u3l4gl.html (explanation of right triangle terminology and practice problems) http://www.quia.com/pop/37541.html (rational & irrational numbers - practice identifying rational and irrational #s) 2
II. Algebraic Expressions and Linear Equations (10 days) Technology: Excel & graphing calculator Translating Phrases, Sentences A.A.3 a. Verbal Phrase to Algebraic Expression A.A.1 b. Write a Verbal Phrase given an Algebraic Expression A.A.2 c. Verbal Sentence to Algebraic Equation A.A.4 Solving Linear Equations A.A.21 & A.A.22 & A.A.25 Evaluate a. Check whether a given number is a solution to an equation or inequality b. Two step equations c. Variables on both sides d. Combining like terms e. Parenthesis (Distributive Property) f. Fractions/Decimals including coefficients g. Fractional equations by clearing the denominator Word Problems A.A.5 & A.A.6 a. Larger/Smaller Numbers b. Consecutive Integers Include even and odd consecutive integers c. Perimeter Problems d. Angles of a Triangle Literal Equations A.A.23 a. Including Formulas: Area, Perimeter, Circumference, Volume, Celsius Fahrenheit, etc. Coefficient Consecutive Constant Decrease Difference Equation Expression Increase Inverse operation Like terms Literal Equation Phrase Product Quantity Quotient Sentence Simplify Solve Statement Sum Translate Twice Variables A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) A.A.1 Translate a quantitative verbal phrase into an algebraic expression A.A.2 Write a verbal expression that matches a given mathematical expression A.A.3 Distinguish the difference between an algebraic expression and an algebraic equation A.A.4 Translate verbal sentences into mathematical equations or inequalities A.A.5 Write algebraic equations or inequalities that represent a situation A.A.21Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable A.A.22 Solve all types of linear equations in one variable A.A.23 Solve literal equations for a given variable A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable. Integrated Mathematics Course 1 Textbook; Chapter 3-1, 3-2, 3-4, 3-5, 4, 10 Algebra 1 Concepts and Skills Textbook; Chapters 1.5, 3, http://www.math.com/school/subject2/lessons/s2u1l3gl.html (words to math - explanation and interactive problems) ; http://www.mathgoodies.com/lessons/vol7/equations.html (equations) www.quia.com/cb/62631.html (Algebra Jeopardy) 3
III. Inequalities (4 days) d Linear Inequalities a. Translating verbal sentences into inequalities A.A.4 b. Write algebraic inequalities that represent a situation A.A.5 c. Determine if a given value is a solution to an inequality A.A.21 d. Solve Linear Inequalities in one variable A.A.24 e. Graph solution on a number line A.G.6 f. Word Problems A.A.6 Compound Inequalities a. Translate verbal sentences into compound inequalities b. Write compound inequalities that represent a situation Graphing using Interval Notation Example Interval Notation: [ 3,12) means -3 x < 12 Graphing using Set Builder Notation Example Set Builder Notation: {x -3 x < 12} < less than < less than or equal to > greater than > greater than or equal to At least At most Closed Circle Compound Inequality Exceeds Inequality Symbols Interval Notation Linear Inequality Maximum of Minimum of Open Circle Set Builder Notation Solution Set A.A.4 Translate verbal sentences into mathematical equations or inequalities A.A.5 Write algebraic equations or inequalities that represent a situation A.A.6 Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable A.A.21Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable A.A.24 Solve linear inequalities in one variable A.A.29 Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form A.G.6 Graph linear inequalities Integrated Mathematics Course 1 Textbook; Chapters 10-10, 18-5 Algebra 1 Concepts and Skills Textbook; Chapters 6.1, 6.2, 6.3, 6.4, 6.8, 7.6 Technology: Graphing Calculator 4
IV. Exponents and Monomials (9 days) Technology: graphing calculator Monomials (add, subtract, combine like terms) A.A.13 Multiplication Properties of Exponents A.A.12 (including power to a power) Zero and Negative Exponents A.A.12 Division Properties of Exponents A.A.12 (including quotient to a power) Scientific notation with Multiplication and Division A.N.4 Problem solving (ex: area, perimeter) A.N.6 Base Common Degree Exponent Exponential Form Leading Coefficient Like Terms Monomial Multiple Negative Exponents Polynomial Power Product to a Power Quotient to a Power Reciprocal Scientific Notation Zero Power A.N.4 Understand and use scientific notation to compute products and quotients of numbers A.N.6 Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s) A.A.12 Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only. A.A.13 Add, subtract, and multiply monomials and polynomials Integrated Mathematics Course 1 Textbook; Chapters 8, 9 Algebra 1 Concepts and Skills Textbook; Chapter 10 Manipulatives: Algebra Tiles 5
V. Polynomials (6 days) Manipulative: algebra tiles Technology: virtual tiles Adding polynomials A.A.13 Subtracting polynomials A.A.13 Multiplying a polynomial by a monomial A.A.13 Multiplying a polynomial by a polynomial A.A.13 Dividing a polynomial by a monomial A.A.14 Base Common Binomial Degree Exponent Exponential Form Leading Coefficient Like Terms Monomial Multiple Negative Exponents Polynomial Power Standard Form Trinomial A.A.13 Add, subtract, and multiply monomials and polynomials A.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainder Integrated Mathematics Course 1 Textbook; Chapters 8, 9 Algebra 1 Concepts and Skills Textbook; Chapter 10 Manipulatives: Algebra Tiles 6
VI. Factoring (7 days) Manipulative: algebra tiles Prime factorization GCF (greatest common factor) A.A.20 DOTS (Difference of two squares) A.A.19 Trinomial with coefficient of 1 A.A.20 Trinomial with coefficient greater than 1 Factor Completely A.A.19, A.A.20 a. GCF with TRI b. GCF with DOTS c. DOTS with DOTS A.A.20 Binomial Composite Degree of a Monomial Degree of a Polynomial Difference of Two Perfect Squares (DOTS) Expression Factor a Polynomial Completely Factor a Trinomial Factored Form Factoring Greatest Common Factor (GCF) Integral Coefficients Leading Coefficient Monomial Perfect Square Trinomial Polynomial Prime Prime Factorization Prime Polynomial Quadratic Trinomial Standard Form Trinomial A.A.19 Identify and factor the difference of two perfect squares A.A.20 Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF) Integrated Mathematics Course 1 Textbook; Chapters 13 Algebra 1 Concepts and Skills Textbook; Chapters 10.5, 10.7, 10.8 http://amby.com/educate/math/2-1_fact.html (prime factorization - explanation and practice problems) www.algebrahelp.com ; www.webmath.com ; www.explorelearning.com 7
VII. Solving Quadratic Equations by Factoring (6 days) Technology: graphing calculator Solving quadratic in factored form A.A.27 Standard form of quadratic equation Solving quadratic equations by factoring A.A.28 Problem solving with quadratics A.A.8 a. Area b. Smaller/Larger Numbers c. Consecutive Integers Binomial Equation Integral root(s) Monomial Quadratic Equation Root(s) of an equation Solution Set Standard Form of a Quadratic Equation Translate (from verbal to symbolic) Trinomial Verbal Expression Zero-Product Property A.A.8 Analyze and solve verbal problems that involve quadratic equations A.A.27 Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots A.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression Integrated Mathematics Course 1 Textbook; Chapters 20-1, 20-2, 20-3, 20-6 Algebra 1 Concepts and Skills Textbook; Chapters 10.4 www.algebrahelp.com ; www.webmath.com ; www.explorelearning.com 8
VIII. Ratio, Proportion, Percent (11 ( days) Ratio & rate (including unit rate) A.M.1 Central Angle Equations with Ratios A.N.5 Conversion a. Ratio of Angles in a Triangle Commission. b. Splitting Profits Commission Rate Solving Proportions A.N.5 Discount a. Linear Equations Estimate Gratuity b. Quadratic Equations Interest Rate c. Scale Drawing & Models Mark-up d. Height and Shadow Measurements e. Similar Figures Percent Conversions Percent Error a. Conversions with Measurement Systems Given Percent of Change (Increase and Decrease) Relationship between Units A.M.2 Principle b. Percent Error in Measurement A.M.3 Proportion Percents Applications Rate a. Part/whole 8.N.4, A.N.5 Ratio b. Discount 8.N.4,, A.N.5 Ratio Notation c. Sale price & sales tax 8.N.4,, A.N.5 Sales Price d. Percent of change 8.N.4,, A.N.5 Sales Tax Scale Drawing e. Simple interest 8.N.4 Scale Model f. Commission 8.N.4 Sector g. Gratuity 8.N.4, 8.N.5, 8.N.6 Similar Figures h. Circle graphs (analyze and construct) 8.CM.4 Simple Interest Unit Rate 8.CM.4: Share organized mathematical ideas through the manipulation of objects, numerical Unit Analysis tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form 8.N.3: Read, write, and identify percents less than 1% and greater than 100% 8.N.4: Apply percents to: tax, percent increase/decrease, simple interest, sale price, commission, interest rates, and gratuities 8.N.5: Estimate a percent of a quantity, given an application 8.N.6: Justify the reasonableness of answers using estimation A.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation A.M.1 Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail) A.M.2 Solve problems involving conversions within measurement systems, given the relationship between the units A.M.3 Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure Integrated Mathematics Course 1 Textbook; Chapters 11-6, 12-1, 12-2, 12-3, 12-4, 12-6 Algebra 1 Concepts and Skills Textbook; Chapters 3.9, 11.1 http://www.math.com/school/subject1/lessons/s1u2l3gl.html (distance, rate and time) http://www.hostsrv.com/webmab/app1/msp/quickmath/02/pagegenerate?site=mathcom&s1=numbers&s2 =percentages&s3=basic (percent practice) www.321know.com ; www.explorelearning.com 9
IX. Algebraic Fractions/Rational F Expressions (8 days) Undefined A.A.15 Simplifying Algebraic Fractions A.A.16,, A.A.14 a. Factor and simplify with GCF b. Factor and simplify with DOTS or TRI Multiplying Algebraic Fractions A.A.18 Dividing Algebraic Fractions A.A.18,, A.A.14 Adding/Subtracting Algebraic Fractions with Like Denominators a. Monomial and Binomial Denominators A.A.17 Adding/Subtracting Algebraic Fractions with Unlike Denominators a. Monomial Only A.A.17 Solving Fractional Equations (Revisit from Previous Unit) A.A.25 & A.A.26 a. Algebraic Proportions b. Linear Equations c. Quadratic Equations Word Problems a. Area b. Perimeter c. Fractional Word Problems (ex: The denominator of a fraction exceeds the numerator by 5. If three is added to both the numerator and denominator, the result is 6/11. Find the original fraction.) Algebraic Fraction Denominator Extremes Fractional Equation Fractional Expression Least Common Denominator (LCD) Lowest terms Means Numerator Proportions Rational Equations Rational Expressions Rational Numbers Undefined A.A.14 Divide a polynomial by a monomial or binomial, where the quotient has no remainder A.A.15 Find values of a variable for which an algebraic fraction is undefined A.A.16 Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms A.A.17 Add or subtract fractional expressions with monomial or like binomial denominators A.A.18 Multiply and divide algebraic fractions and express the product or quotient in simplest form A.A.25 Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable. A.A.26 Solve algebraic proportions in one variable which result in linear or quadratic equations Integrated Mathematics Course 1 Textbook; Chapters 14 Algebra 1 Concepts and Skills Textbook; Chapters 11.3, 11.4, 11.5, 11.6,11.7 10
X. Plane & Solid Geometry (7( days) Manipulative: geoboards Technology: Geometer s Sketchpad Pairs of Angles a. Identify pairs of vertical angles as congruent (5-1) 8.G.1 b. Calculate missing angle measurements from intersecting lines (5-1, 5-2) 8.G.6 c. Identify pairs of supplementary and complementary angles (5-1) 8.G.2 d. Calculate the missing angle in a supplementary or complementary pair (5-1) 8.G.3 e. Apply algebra to determine missing measures of above angles 8.A.12 Parallel Lines Cut by a Transversal a. Determine angle pair relationships (5-2) 8.G.4 b. Calculate missing angle measurements (5-2) 8.G.5 c. Apply algebra to determine the measure of above angles 8.A.12 Area and Perimeter of Figures A.G.1 a. Composition of figures (See Performance Indicators below) b. Shaded Region Volume A.G.2 a. Rectangular Solids and Cylinders Surface Area A.G.2 a. Rectangular Solids and Cylinders Adjacent Angles Alternate Interior Angles Alternate Exterior Angles Area Circle Circumference Complementary Angles Congruent Angles Corresponding Angles Cylinder Formula Geometric Shape Intersecting Angles Linear Pair Parallel Lines Parallelogram Perimeter Perpendicular Lines Polygon Quarter Circle Rectangle Rectangular Solid Regular Polygon Rhombus Same Side Interior Angles Sector Semi-Circle Square Surface Area Supplementary Angles Transversal Trapezoid 8.A.12: Apply algebra to determine the measure of angles formed by or contained in parallel lines Triangle cut by a Vertical Angles Volume transversal and by intersecting lines 8.G.1: Identify pairs of angles as congruent 8.G.2: Identify pairs of angles as supplementary and complementary angles 8.G.3: Calculate the missing angles in a supplementary or complementary pair 8.G.4: Determine angle pair relationship when given two parallel lines cut by a transversal 8.G.5: Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6: Calculate the missing angle measurements when given two intersecting lines and an angle A.G.1 Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons (perimeter only). A.G.2 Use formulas to calculate volume and surface area of rectangular solids and cylinders Integrated Mathematics Course 1 Textbook; Chapters 3-6, 19-13, 19-14 11
XI. Transformational Geometry (Take Home Packet) Technology: Geometer s Sketchpad Describe and identify transformations in the coordinate plane, using proper function notation 8.G.7 a. Translations (5-7) b. Reflections (5-7) c. Dilations (7-5) d. Rotations (5-7) e. Symmetry - Line and Rotational (5-8) Draw the image of the following a. A figure under rotations of 90 o and 180 o (5-7) 8.G.8 b. A figure under a reflection of a given line (5-7) 8.G.9 c. A figure under a translation (5-7) 8.G.10 d. A figure under a dilation (7-5) 8.G.11 Identify the properties preserved and not preserved under each of the above (5-7, 7-5) 8.G.12 Center of Rotation Coordinate Place Dilation Enlarge Flip Image (Prime Notation) Line of Reflection Line of Rotation Line of Symmetry Line Symmetry Mapping Mirror Image Not Preserved Pre-Image Preserved Reduce Reflection Rotation Rotational Symmetry Similarity Slide Translation Transformation 8.G.7: Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations 8.G.8: Draw the image of a figure under rotations of 90 and 180 degrees 8.G.9: Draw the image of a figure under a reflection over a given line 8.G.10: Draw the image of a figure under a translation 8.G.11: Draw the image of a figure under a dilation 8.G.12: Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation Technology: Geometer s Sketchpad Internet: www.explorelearning.com; www.mathsisfun.com ; www.mathsonline.co.nz 12
XII.. Functions (10 Days) Describe a situation involving relationships that matches a given graph (1-9) 8.A.3 Create a graph given a description for a situation involving a linear or non-linear relationships 8.A.4 Evaluate functions using function notation (ex: f(3) implies when x = 3) Identify Domain and Range of Functions 8.A.17 Represent patterns and sequences in multiple ways (diagrams, tables, graphs) (12-1, 12-2, 12-3, Lab 12A, Tech lab p635) 8.A.15,, 8.A.19 Identify and graph linear, quadratic, absolute value and exponential functions using a table of values A.G.4,, 8.A.16 Determine when a relation is a function by examining ordered pairs A.G.3,, 8.A.18 Determine when a relation is a function by its graph using the vertical line test A.G.3,, 8.A.18 Investigate how changing the coefficients of linear, quadratic, absolute value and exponential functions affects the graph A.G.5 a. Positive and Negative Coefficients Exponential Growth and Decay A.A.9 a. Ex: Interest problems; A = P(1 + r) t, (no compounding) 8.A.3: Describe a situation involving relationships that matches a given graph Absolute Value Function Arithmetic Sequence Coefficients Common Difference Common Ratio Domain Exponential Function Exponential Growth/Decay Fibonacci Sequence Function Function Notation Geometric Sequence Input Linear Functions Multiple Representations Ordered Pairs Output Parabola Quadratic Function Non-Linear Range Relation Sequence Term Vertical Line Test 8.A.4: Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship 8.A.15: Understand that numerical information can be represented in multiple ways: arithmetically, algebraically, and graphically 8.A.16: Find a set of ordered pairs to satisfy a given linear numerical pattern (expressed algebraically); then plot the ordered pairs and draw the line 8.A.17: Define and use correct terminology when referring to function (domain and range) 8.A.18: Determine if a relation is a function 8.A.19: Interpret multiple representations using equation, table of values and graph A.A.9 Analyze and solve verbal problems that involve exponential growth and decay A.G.3 Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations A.G.4 Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions A.G.5 Investigate and generalize how changing the coefficients of a function affects its graph Algebra 1 Concepts and Skills Textbook; Chapters 1.8, 4.8, 8.3, 8.7 Technology: Graphing Calculator http://illuminations.nctm.org/activitydetail.aspx?id=38 (spreadsheet and graphing calculator) ; http://illuminations.nctm.org/activitydetail.aspx?id=110 (trout population calculator) 13
XIII III. Radicals (7 days) Radicals a. Rational vs. Irrational Numbers b. Simplify Radicals A.N.2 c. Add and Subtract Radicals A.N.3 Like and Unlike Radicands d. Multiply and Divide Radicals A.N.3 Pythagorean Theorem A.A.45 Coefficients Converse Hypotenuse Irrational Leg Perfect Square Radical Radicands Rational Right Triangle Square Root a. Find legs and hypotenuse b. Express answers in simplest radical form c. Round answers to indicated place value d. Determine whether three given sides form a right triangle (Converse) A.N.2 Simplify radical terms (no variable in the radicand) A.N.3 Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form A.A.45 Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides Integrated Mathematics Course 1 Textbook; Chapters 1, 7, 19, 20-5 Algebra 1 Concepts and Skills Textbook; Chapters 2, 9, 12.6 http://www.math.com/school/subject3/lessons/s3u3l4gl.html (explanation of right triangle terminology and practice problems) http://www.quia.com/pop/37541.html (rational & irrational numbers - practice identifying rational and irrational #s) 14
XIV. Right Triangle Trigonometry (5 days) Supplemental packet Manipulative: clinometer Technology: Geometer s Sketchpad Trig ratios (sine, cosine, tangent) A.A.42 Finding missing sides (using both chart & calculator) A.A.44 Finding missing angles (using both chart & calculator) A.A.43 Problem solving a. Angle of Elevation and Depression Optional lab (clinometer activity) Adjacent Side Angle of Depression Angle of Elevation Hypotenuse Legs of a Right Triangle Opposite Side Ratio Sine Cosine Tangent Trigonometry A.A.42 Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides A.A.43 Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle A.A.44 Find the measure of a side of a right triangle, given an acute angle and the length of another side Integrated Mathematics Course 2 Textbook; Chapters 8 Technology: Graphing Calculator 15
XV. Graphing Linear Equations (7 days) Technology: Excel & graphing calculator Graphing vertical and horizontal lines Determine if a point lies on a line A.A.39 Graphing linear equations using x and y intercepts Slope of a line a. Definition of a slope b. Slope as a rate of change (positive, negative, zero and undefined) A.A.32 c. Finding the slope using a graph A.A.33 d. Finding the slope using two points with the slope formula A.A.33 e. Determine the slope of a line given its equation in any form A.A.37 Direct variation (y = kx) where k is the Constant of Variation A.N.5 Graphing linear equations using slope intercept form Constant Function Constant of Variation Coordinates Dependent Variable Direct Variation Graph of an Equation Independent Variable Integral Coefficient Linear Equation Negative Slope No Slope Ordered Pair Origin Parallel Lines Positive Slope Quadrant Rational Coefficient Slope Slope-Intercept Form Undefined Slope x-axis x-coordinate x-intercept y-axis y-coordinate y-intercept Zero Slope A.A.32 Explain slope as a rate of change between dependent and independent variables A.A.33 Determine the slope of a line, given the coordinates of two points on the line A.A.37 Determine the slope of a line, given its equation in any form A.A.39 Determine whether a given point is on a line, given the equation of the line A.N.5 Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation 16
XVI. Writing Linear Equations (6 days) Technology: Excel & graphing calculator Writing linear equations given slope and y intercept Writing linear equations given slope and point A.A.34 Writing linear equations given two points A.A.35 Writing linear equations of lines parallel to the x or y-axis (vertical and horizontal lines) A.A.36 Writing linear equations of parallel lines A.A.38 Writing equations for linear models Coordinates Graph of an Equation Linear Equation Negative Slope No Slope Ordered Pair Origin Parallel Lines Positive Slope Quadrant Slope Slope-Intercept Form Undefined Slope x-axis x-coordinate x-intercept y-axis y-coordinate y-intercept Zero Slope A.A.34 Write the equation of a line, given its slope and the coordinates of a point on the line A.A.35 Write the equation of a line, given the coordinates of two points on the line A.A.36 Write the equation of a line parallel to the x- or y-axis A.A.38 Determine if two lines are parallel, given their equations in any form Integrated Mathematics Course 1 Textbook; Chapters 12-5, 17 Algebra 1 Concepts and Skills Textbook; Chapters 5.1, 5.3, 5.4 Technology: Graphing Calculator www.mrhudak.com/powerpoint.html 17
XVII. Systems of Equations and Inequalities (8 days) Technology: Excel & graphing calculator Solving linear systems by graphing a. Include Word Problems (ex: Initial Cost, Monthly Fee, Which is a better buy? When are they the same?) A.A.7 Solving linear systems by substitution A.A.10 Solving linear systems by addition (linear combinations) A.A.10 a. Multiplying one equation by a constant b. Multiplying both equations by a constant Applications of linear systems A.A.7 Special types of linear systems (infinite & no solution) A.A.10 Graphing linear inequalities on coordinate plane A.G.6 Performance Solving and Indicators: graphing systems of linear inequalities A.G.7 A.G.6 Determine Graph linear if inequalities a point is a solution of a system of inequalities A.A.7 A.A.40 Analyze and solve verbal problems whose solution requires solving systems of Boundary linear Line equations in two variables A.A.10 Solve systems of two linear equations in two variables algebraically A.A.40 Determine whether a given point is in the solution set of a system of linear inequalities A.G.7 Graph and solve systems of linear equations and inequalities with rational coefficients in two variables Better buy, deal Elimination Infinite, Many Solutions No Solution Point of Intersection Substitution System of Equations System of Inequalities Integrated Mathematics Course 1 Textbook; Chapters 12-5, 17 Algebra 1 Concepts and Skills Textbook; Chapters 5.1, 5.3, 5.4 Technology: Graphing Calculator 18
XVIII. Graphing Quadratic Equations (8 days) Technology: Graphing Calculator Axis of Symmetry Factor Line of Reflection Maximum Minimum Non-Linear Equation Parabola Point(s) of Intersection Quadratic Equation Quadratic-Linear System Roots Solve Algebraically Solve Graphically Turning Point Vertex Using table of values (graphing calculator) Axis of Symmetry A.A.41, A.G.10 Turning point (vertex) A.A.41.A.41, A.G.10 Maximum and Minimum Find the roots algebraically (factor) and compare to the graph A.A.28 Find the roots graphically (integral values) A.G.8 a. From table/graph b. Using Calc zero on Calculator Applications a. Find Height: Maximum/Minimum b. Hits the ground ; Root c. Time in the air; Root Solving Quadratic-Linear Systems (Integral values only) a. Graphically A.G.9 b. Algebraically (no leading coefficients) A.A.11 A.A.11 Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers. A.A.28 Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression A.A.41 Determine the vertex and axis of symmetry of a parabola, given its equation A.G.8 Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions. A.G.9 Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers. A.G.10 Determine the vertex and axis of symmetry of a parabola, given its graph Note: The vertex will Integrated Mathematics Course 1 Textbook; Chapters 18 have an ordered pair of integers and the axis of symmetry will have an integral value. Algebra 1 Concepts and Skills Textbook; Chapters 7.1, 7.2, 7.3, 7.4, 7.5, 19
Certain Events Complements XIX. Probability (9( days) Compound Event Counting Principle Tree diagram/sample space A.S.19 Dependent Events Counting Principle A.N.7 Empirical Probability Factorials A.N.6 Equally Likely Permutations A.N.8 Event Experimental a. Words with and without repetition Factorial Probability of an event and its complement A.S.20 & A.S.22 Favorable Event Theoretical vs. Experimental Probability Impossible Events a. Find probabilities given sample data A.S.21 Independent Events Probability (single event) A.S.18 Intersection of two a. And events (and) b. Or Mutually Exclusive Not Equally Likely Performance Indicators: Probability (multiple events) A.S.23 Ordered Pairs A.N.6 Evaluate expressions a. With involving Replacement factorial(s), absolute value(s), and exponential expression(s) Outcome A.N.7 Determine the number b. Without of possible Replacement events, using counting techniques or the Fundamental Principle of Counting Permutations A.N.8 Determine the Finding number Probability of possible arrangements of Mutually Exclusive (permutations) Events of A.S.23 a list of items Probability of One A.S.18 Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces a. Example: Rolling a die and flipping a coin Probability of Zero A.S.19 Determine the number of elements in a sample space and the number of favorable events A.S.20 Calculate the Venn probability Diagrams of an event A.A.31 and its complement A.S.21 Determine empirical a. And/Or probabilities questions based on specific sample data Sample Space Theoretical Tree Diagrams Trials Union of two events (or) Venn Diagram A.S.22 Determine, based on calculated probability of a set of events, if: o some or all are equally likely to occur o one is more likely to occur than another o whether or not an event is certain to happen or not to happen A.S.23 Calculate the probability of: o a series of independent events o a series of dependent events o two mutually exclusive events o two events that are not mutually exclusive A.A.31 Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets) Integrated Mathematics Course 1 Textbook; Chapters 20-7 Integrated Mathematics Course 2 Textbook; Chapters 10-7, 10-8, 10-11, 10-12 Algebra 1 Concepts and Skills Text Book; Chapters 9.4, 9.5 Technology: Graphing Calculator 20
XX. Statistics (10 Days) 5 number summary Appropriateness Bar Graph Biased Bi-modal Bivariate Box and whisker Causation Census Central Angle Central Tendency Circle Graph Correlation Cumulative Frequency Data Extrapolation Frequency Table Histogram Interpolation Interquartile range Interval Line of best fit Maximum Mean Median Minimum Misleading Statistics Modal Interval Mode Outlier Percentile Population Qualitative Quantitative Quartiles (specifically; first, second, third, or lower, middle, upper) Random Data a. Qualitative (descriptive) vs. Quantitative (numerical) A.S.1 b. Bivariate (2-Variables) vs. Univariate (1-Variable) A.S.2 c. Biased vs. Non-biased A.S.3, A.S.10, A.S.15 d. Outliers Measures of Central Tendency and Range A.S.16 a. Appropriateness of each A.S.4 Construct and Interpret each of the following displays of data a. Frequency Histogram A.S.5 & A.S.9 o Table o Graph o Percentiles and Quartiles A.S.11 b. Cumulative Frequency Histogram A.S.5 & A.S.9 o Table o Graph o Percentiles and Quartiles A.S.11 c. Box and Whisker Plot A.S.5 & A.S.6 & A.S.9 o Lower Quartile A.S.11 o Median A.S.11 o Upper Quartile A.S.11 o Minimum value o Maximum value d. Scatter Plot A.S.7 o Independent vs. Dependent o Line of Best Fit (graphing calculator) A.S.8 Make Predictions based on the line of best fit A.S.17 o Correlation A.S.12 Positive Integrated Mathematics Course Negative 1 Textbook; Chapters 15 Zero Technology: Graphing Calculator Correlation vs. Causation A.S.13 & A.S.14 Random Sample Internet: www.regentsprep.org; www.jmap.org ; http://illuminations.nctm.org ; www.unitedstreaming.com Range http://illuminations.nctm.org/activitydetail.aspx?id=67 (random drawing tool individual trials) Sample ; http://illuminations.nctm.org/activitydetail.aspx?id=159 (random drawing tool sampling distribution) Scale ; http://illuminations.nctm.org/activitydetail.aspx?id=79 (adjustable spinner) Scatter plot 21 Tally Unbiased Uni-variate
A.S.1 Categorize data as qualitative or quantitative A.S.2 Determine whether the data to be analyzed is univariate or bivariate A.S.3 Determine when collected data or display of data may be biased (Performance Indicators and Resources for this unit are on following page) A.S.4 Compare and contrast the appropriateness of different measures of central tendency for a given data set A.S.5 Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data A.S.6 Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plot A.S.7 Create a scatter plot of bivariate data A.S.8 Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line A.S.9 Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot A.S.10 Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions A.S.11 Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles A.S.12 Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none) A.S.13 Understand the difference between correlation and causation A.S.14 Identify variables that might have a correlation but not a causal relationship A.S.15 Identify and describe sources of bias and its effect, drawing conclusions from data A.S.16 Recognize how linear transformations of one-variable data affect the data s mean, median, mode, and range A.S.17 Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation Integrated Mathematics Course 1 Textbook; Chapters 16 Technology: Graphing Calculator, Excel http://nces.ed.gov/nceskids/graphing/ (graphing scatter plots) ; http://illuminations.nctm.org/activitydetail.aspx?id=160 (mean, median, box-and-whisker plot) ; http://illuminations.nctm.org/activitydetail.aspx?id=82 (line of best fit) ; http://illuminations.nctm.org/activitydetail.aspx?id=77 (box and whisker plot) ; http://illuminations.nctm.org/activitydetail.aspx?id=146 (line of best fit) 22