Geneva CUSD 304 Content-Area Curriculum Frameworks Grades 6-12 Mathematics Mission Statement The study of mathematics can be an exciting and interesting challenge. Yet, the major reason to become proficient in this discipline revolves around the use of mathematics as a tool to solve from the areas of science, business, engineering, economics, and all other areas which involve data collection and analysis. The mathematics program is designed to establish connections between the key concepts of math and the applications. Theoretical structure of mathematics - Students will demonstrate an understanding of the theoretical foundations of mathematics. Thought processes (intuition, deduction, induction) Logical arguments (two column, narrative, flow chart proofs) Structure of axiomatic systems (Euclidian geometry, real and complex number systems) Fundamental concepts (functions, sets, limits, infinity, infinitesimals, statistics, probability) Problem solving - Students will formulate problem solving strategies. Establishment of relationships (numerical, geometric, pictorial, graphic, symbolic) Recognition, collection, and analysis of pertinent data Development and evaluation of methods and algorithms Validation of results (estimation, approximation, reasonableness) Mechanics of mathematics - Students will symbolically manipulate mathematical expressions and statements. Performance of operations and computational processes (arithmetic, algebraic, graphic) Illustration of solution processes for equations and inequalities Calculation using electronic devices (scientific and graphing) Algebra I A_B Frameworks.doc Page 1 of 16
Appropriate use of emerging technology - Students will use technology to improve and extend their understanding of mathematics. Calculators at appropriate levels (scientific and graphing) Computer software (graphic, spread sheets, data bases, symbolic manipulators, simulations) Information management systems (compact disk, telecommunication, internet, video disk) All students will experience an evolving curriculum designed to be a rich tapestry of traditional mathematics skills intertwined with problem solving, graphical analysis, measurement, probability, and statistics. The use of manipulatives, calculators (scientific and graphing), computers, writing assignments, and cooperative learning activities will all be designed to achieve this mission. Course Sequence (Grades 6-12) 6 th Grade Mathematics 7 th Grade Mathematics Pre- Algebra Integrated Mathematics I, II Algebra I (4 semesters) A and B Algebra I Geometry /Concepts and Applications Geometry (Regular & Honors) Algebra II (Regular & Honors) Algebra II 1/3-3/3 Pre-Calculus (Regular & Honors) Trigonometry (Regular & Honors) Calculus AP Calculus AP Statistics Algebra I A_B Frameworks.doc Page 2 of 16
Course Framework Course Title Grade Level Semesters (1-2-3-4) Prerequisite Course Description District-approved Materials and/or Resources Algebra I A/B 9, 10, 11, 12 4 None Algebra I - Part A and Algebra I - Part B (4 semesters) gives students a foundation for future mathematics, business, and science courses. Algebra I (4 semesters) provides an introduction to the basic algebra skills needed in problem solving. The use of calculators (scientific and graphing) is integrated throughout the course to provide students with a realistic view of how technology can be used as a tool for approximating answers, solving and validating the results. The graphing calculator is a tool that the student may use on the Prairie State Achievement Exam. Proficiency on this calculator will be an asset for the student. The content of this course is equivalent to the Algebra I course. This course is taught over four semesters rather than two, thereby providing additional time for skill development and process understanding. This course provides the prerequisite for Geometry Concepts & Applications and Geometry. Algebra I McDougal Littell Larson Boswell, Kanold and Stiff ISBN# 0-395-97888-2 Algebra I A_B Frameworks.doc Page 3 of 16
Unit of Study: Unit Frameworks Connections to Algebra and The Real Number System Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit 6.A.4 Identify and apply the associative, commutative, distributive and identity properties of real numbers, including special numbers such as pi and square roots. 6.B.4 Select and use appropriate arithmetic operations in practical situations including calculating wages after taxes, developing a budget and balancing a checkbook. 6.C.4 Determine whether exact values or approximations are appropriate (e.g., bid a job, determine gas mileage for a trip). 6.D.4 Solve involving recipes or mixtures, financial calculations and geometric similarity using ratios, proportions and percents. 7.A.4a Apply units and scales to describe and compare numerical data and physical objects. 7.A.4b Apply formulas in a wide variety of theoretical and practical realworld measurement applications involving perimeter, area, volume, angle, time, temperature, mass, speed, distance, density and monetary values. Objectives o Conceptual o Factual o Procedural 7.C.4c Convert within and between measurement systems and monetary systems using technology where appropriate 1. Evaluate a variable expressions 2. Write a variable expression that models a real-life situation 3. Evaluate expressions containing exponents 4. Use exponents in real-life 5. Use the order of operations to evaluate algebraic expressions 6. Check to determine if given values are solutions to specified equations and inequalities 7. Translate verbal phrases into algebraic expressions Algebra I A_B Frameworks.doc Page 4 of 16
8. Use a verbal model to write an algebraic equation or inequality to solve real-life 9. Use tables and graphs to organize real life data 10. Identify a function and make an input-output table for a function 11. Write an equation for a real-life function 12. Graph and compare real numbers using a number line 13. Compute the absolute value of a number. 14. Add and subtract real numbers in rote and real-life 15. Organize data in a matrix 16. Add and subtract two matrices 17. Multiply and divide real numbers in rote and real-life 18. Use the distributive property 19. Simplify expressions by distributing and combining like terms 20. Find the probability and odds of an event Assessments Performance tasks, chapter quizzes, chapter tests, and semester exams will be given. They will include multiple choice and /or free response. Other Evidence: Homework Board/Class Work Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 5 of 16
Unit Frameworks Unit of Study: Linear Equations and Inequalities Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit 6.A.4 Identify and apply the associative, commutative, distributive and identity properties of real numbers, including special numbers such as pi and square roots 6.D.4 Solve involving recipes or mixtures, financial calculations and geometric similarity using ratios, proportions and percents. 8.A.4b Represent mathematical patterns and describe their properties using variables and mathematical symbols 8.B.4a Represent algebraic concepts with physical materials, words, diagrams, tables, graphs, equations and inequalities and use appropriate technology. 8.B.4b Use the basic functions of absolute value, square root, linear, quadratic and step to describe numerical relationships. 8.C.4b Apply algebraic properties and procedures with matrices, vectors, functions and sequences using data found in business, industry and consumer situations. 8.D.4 Formulate and solve linear and quadratic equations and linear inequalities algebraically and investigate nonlinear inequalities using graphs, tables, calculators and computers. 10.A.4a Represent and organize data by creating lists, charts, tables, frequency distributions, graphs, scatter plots and box-plots. 10.A.4b Analyze data using mean, median, mode, range, variance and standard deviation of a data set, with and without the use of technology. 10.A.4c Predict from data using interpolation, extrapolation and trend lines, with and without the use of technology. 10.C.4a Solve of chance using the principles of probability including conditional settings. Algebra I A_B Frameworks.doc Page 6 of 16
Objectives o Conceptual o Factual o Procedural 1. Solve one-step linear equations using addition, subtraction, multiplication and division 2. Solve multi-step linear equations 3. Solve linear equations with variables on both sides 4. Find exact and approximate solutions of equations that contain decimals 5. Use linear equations, diagrams, and tables to model and solve real-life 6. Solve a formula or literal equation for one of its variables 7. Rewrite and equation in function form 8. Use rates, ratios, and percents to model and solve real-life 9. Plot points in a coordinate plane 10. Draw a scatter plot and make predictions about real-life situations 11. Graph a linear equation using a table of values 12. Graph vertical and horizontal lines 13. Find the x and y intercepts of a linear equation and use these to make graphs of linear equations 14. Interpret slope as a rate of change in real-life situations 15. Find the slope of a line using two of its points 16. Write linear equations that represent direct variations in rote and reallife 17. Graph and interpret a linear equation in slope intercept form in both rote and real-life 18. Identify when a relation is a function 19. Use function notation to evaluate functions in rote and real-life 20. Identify equations of lines in point-slope, slope-intercept, and general Algebra I A_B Frameworks.doc Page 7 of 16
form 21. Convert linear equations from point-slope to slope-intercept and general forms 21. Write the equation of a line given a slope and a y-intercept in rote and real-life 22. Write the equation of a line given a slope and a point in rote and reallife 23. Write the equation of a line given two points in rote and real-life 24. Approximate equations for lines of best fit using real-life data and use these equations to make predictions 25. Graph linear inequalities in one variable 26. Solve one and multi-step linear inequalities and graph the solution on a number line in rote and real-life 27. Write, solve, and graph compound inequalities in rote and real-life 28. Solve absolute value equations and inequalities in rote and real-life 29. Graph a linear inequality in two variables 30. Model a real-life situation using a linear inequality in two variables 31. Find mean, median, and mode of data sets 32 Make stem-and-leaf and box-and whisker plots to analyze data in rote and real-life 33. Solve a system of linear equations by graphing in rote and real-life 34. Use substitution to solve a system of linear equations in rote and reallife 35. Use linear combination (elimination) to solve a system of linear equations in rote and real-life Algebra I A_B Frameworks.doc Page 8 of 16
36. Determine the most efficient method to solve a given system of linear equations 37. Recognize and solve systems of linear equations having one solution, no solution, or infinitely many solutions 38. Graph systems of linear inequalities. Assessments 39. Use a system of linear inequalities to model a real-life situation Performance tasks, chapter quizzes, Other Evidence: chapter tests, and semester exams will be given. They will include multiple Homework choice and /or free response. Board/Class Work Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 9 of 16
Unit Frameworks Unit of Study: Exponents Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural 6.B.5 Identify, represent and apply numbers expressed in exponential, logarithmic and scientific notation using contemporary technology. 6.D.5 Solve involving loans, mortgages and other practical applications involving geometric patterns of growth. 8.B.5 Use functions including exponential, polynomial, rational, parametric, logarithmic, and trigonometric to describe numerical relationships. 8.C.5 Use polynomial, exponential, logarithmic and trigonometric functions to model situations. 8.D.5 Formulate and solve nonlinear equations and systems including involving inverse variation and exponential and logarithmic growth and decay. 1. Use properties of exponents to multiply exponential expressions 2. Use powers to model real-life 3. Evaluate powers that have zero and negative exponents 4. Graph exponential functions 5. Use the division properties of exponents to evaluate powers and simplify expressions 6. Use scientific notation to represent numbers and perform operations with them. 7. Write, use, and graph models for exponential growth and decay in rote and real-life Assessments Performance tasks, chapter quizzes, chapter tests, and semester exams will be given. They will include multiple choice and /or free response. Other Evidence: Homework Board/Class Work Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 10 of 16
Unit Frameworks Unit of Study: Polynomials Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit 6.D.4 Solve involving recipes or mixtures, financial calculations and geometric similarity using ratios, proportions and percents. 7.A.4b Apply formulas in a wide variety of theoretical and practical realworld measurement applications involving perimeter, area, volume, angle, time, temperature, mass, speed, distance, density and monetary values. 8.B.5 Use functions including exponential, polynomial, rational, parametric, logarithmic, and trigonometric to describe numerical relationships. 8.B.4b Use the basic functions of absolute value, square root, linear, quadratic and step to describe numerical relationships. 8.C.4a Analyze and report the effects of changing coefficients, exponents and other parameters on functions and their graphs. 8.C.5 Use polynomial, exponential, logarithmic and trigonometric functions to model situations. 8.D.5 Formulate and solve nonlinear equations and systems including involving inverse variation and exponential and logarithmic growth and decay. Objectives o Conceptual o Factual o Procedural 1. Evaluate and approximate square roots 2. Solve quadratic equations by finding square roots in rote and real-life. 3. Use properties of radicals to simplify radicals 4. Sketch the graph of a quadratic function 5. Solve quadratic equations graphically 6. Solve quadratic equations using the quadratic formula 7. Use quadratic models to in real-life applications 8. Use the discriminate to determine the number of solutions a quadratic Algebra I A_B Frameworks.doc Page 11 of 16
equation will have 9. Sketch the graph of a quadratic inequality in two variables 10. Use quadratic inequalities as real-life models 11. Select the best model to represent a given data set 12. Add and subtract polynomials 13. Multiply two or more polynomials 14. Use difference of two squares and perfect square product patterns 15. Use polynomials to model real-life situations 16. Solve polynomial equations in factored form. 17. Relate factors and x-intercepts 18. Factor quadratic expressions of the form 19. Factor quadratic expressions of the form 2 x + bx + c 2 ax + bx + c 20. Factor quadratic expressions using the difference of two squares product pattern 21. Factor quadratic expressions using the perfect square trinomial product pattern 22. Factor polynomial expressions using the grouping method Assessments 23. Solve quadratic equations by factoring in rote and real-life Performance tasks, chapter quizzes, Other Evidence: chapter tests, and semester exams will be given. They will include multiple Homework choice and /or free response. Board/Class Work Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 12 of 16
Unit Frameworks Unit of Study: Rational Equations and Functions Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit Objectives o Conceptual o Factual o Procedural 6.D.4 Solve involving recipes or mixtures, financial calculations and geometric similarity using ratios, proportions and percents. 8.A.4b Represent mathematical patterns and describe their properties using variables and mathematical symbols. 8.B.4b Use the basic functions of absolute value, square root, linear, quadratic and step to describe numerical relationships. 8.C.4b Apply algebraic properties and procedures with matrices, vectors, functions and sequences using data found in business, industry and consumer situations. 1. Solve proportions to solve rote and real-life 2. Use equations to solve rote and real-life percent 3. Use direct and inverse variation in rote and to model real-life situations 4. Simplify rational expressions 5. Use rational expressions to find geometric probability 6. Multiply and divide rational expressions 7. Add and subtract rational expressions 8. Use rational expressions as real-life models 9. Divide a polynomial by a monomial Assessments 10. Solve rational equations in rote and real-life Performance tasks, chapter quizzes, Other Evidence: chapter tests, and semester exams will be given. They will include multiple Homework choice and /or free response. Board/Class Work Algebra I A_B Frameworks.doc Page 13 of 16
Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 14 of 16
Unit Frameworks Unit of Study: Connections to Geometry Resources that will support instruction Illinois Learning Standards, Benchmarks, National Standards Assessment Frameworks, or other standards that will be taught in this unit 7.C.4a Make indirect measurements, including heights and distances, using proportions (e.g., finding the height of a tower by its shadow). 8.B.5 Use functions including exponential, polynomial, rational, parametric, logarithmic, and trigonometric to describe numerical relationships. 8.C.5 Use polynomial, exponential, logarithmic and trigonometric functions to model situations. 9.A.5 Use geometric figures and their properties to solve in the arts, the physical and life sciences and the building trades, with and without the use of technology. 9.B.4 Recognize and apply relationships within and among geometric figures. 9.D.4 Analyze and solve involving triangles (e.g., distances which cannot be measured directly) using trigonometric ratios. Objectives o Conceptual o Factual o Procedural 1. Evaluate and graph a square-root function 2. Solve radical equations 3. Use square root functions and radical equations to solve real-life 4. Use the Pythagorean Theorem and its converse in rote and real-life 5. Apply the Distance Formula in a coordinate plane. 6. Apply the Midpoint Formula in a coordinate plane. 7. Use sine, cosine, and tangent of an angle in rote and real-life Assessments Performance tasks, chapter quizzes, chapter tests, and semester exams will be given. They will include multiple choice and /or free response. Other Evidence: Homework Board/Class Work Algebra I A_B Frameworks.doc Page 15 of 16
Other learning activities that may include: Cooperative Learning Activities Technology Based Discovery Activities Algebra I A_B Frameworks.doc Page 16 of 16