Honors Course Overview Honors Math II On this page you will find resources that will help you understand the difference between a standard and an honors Math II course. You will also find resources that will guide you throughout the semester in understanding the curriculum, instruction, and assessments. Curriculum Content Math II continues a progression of the standards established in Math I which include concepts of algebra, geometry, functions, number and operations, statistics and modeling throughout the course. Included in these concepts are expressions in the real number system, creating and reasoning with equations and inequalities, interpreting and building simple functions, expressing geometric properties and interpreting categorical and quantitative data. In addition to these standards, Math II includes: polynomials, congruence and similarity of figures, trigonometry with triangles, modeling with geometry, probability, making inferences, and justifying conclusions. This course will allow students to develop skills with functions and to recognize properties of graphs of those functions. Students will discover through exploratory activities, projects, group work, and hands on activities. Honors Math II will have more rigorous standards than a standard Math II course, and expectations will be that students will learn the basic standards quickly and will be ready to dig deeper into the content, allowing time for the extensions listed below within the semester course. Students taking Honors Math II are most likely the students who will take Honors Math III, PreCalculus Honors, as well as an AP Course in Calculus or Statistics. These students need to be prepared for these upper level courses while mastering the basics of Math II. Through vertical alignment, our math department has chosen several topics that would benefit everyone as they move through the honors math pathway. These extensions are aligned to standards in Math III, Honors PreCalculus, AP Calculus, as well as ACT and SAT standards. Standards and Objectives 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. Note: Extend to linear-quadratic, and linear inverse variation systems of equations. Solving systems of equations is a large concept that is used from Math II all the way through every math course students could take in high school. One of the methods to solve them was removed from the curriculums for Math I, II, and III that is extremely helpful for PreCalculus and Calculus courses. Using matrices to solve systems of equations is an extension that would benefit all honors students as they move on to higher math courses. A second extension on this same standard is vocabulary. Vocabulary for systems (consistent, dependent, independent, inconsistent) is not a requirement at this level, but would give students a jumpstart for Math III vocabulary and understanding potential questions for the ACT and SAT involving systems of equations. A-REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Note: At this level, limit to inverse variation.
Solving rational equations is also a concept that is carried throughout future math courses. Solving rational equations that are more difficult and requiring students to find common denominators for these types of problems would help them with solving equations in higher math courses. A-REI.4 Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Note: Solve quadratic equations that have real solutions and recognize quadratic equations that do not have a real solution. (Writing complex solutions is not expected in Math II.) There are multiple ways to solve quadratic equations. Math II focuses on factoring, square roots, as well as the quadratic formula. Completing the square to solve quadratics is another method that isn t explored in Math II standards. Since Math II doesn t involve imaginary roots, often, completing the square isn t necessary. However, by extending this standard to include completing the square, we can also connect perfect square trinomials as well as square roots in solving equations. F-IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Note: At this level, extend to simple trigonometric functions (sine, cosine, and tangent in standard position) Amplitude and Period of Sine and Cosine are easy to find even when they are not in parent forms. Explaining Amplitude and Period connects students prior knowledge with vertical stretches and compressions as well as horizontal stretches and compressions. This allows students to understand how trigonometric functions can be transformed in the same ways as the other functions we have discussed in Math II. F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Note: At this level, completing the square is still not expected. There are multiple ways to solve quadratic equations. Math II focuses on factoring, square roots, as well as the quadratic formula. Completing the square to solve quadratics is another method that isn t explored in Math II standards. Since Math II doesn t involve imaginary roots, often, completing the square isn t necessary. However, by extending this standard to include completing the square, we can also connect perfect square trinomials as well as square roots in solving equations. G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. Note: At this level, include measures of interior angles of a triangle sum to 180 and the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. For Math II, formal proofs are not included in the curriculum. However, by introducing either paragraph or two-column proofs, students will have a better grasp of the thought process required for proving something is true in mathematics. Proofs of congruent triangles are not included in Math III, but they are required to do proofs involving other theorems and topics. This would give students a base knowledge on the process of proofs and help them to think through the requirements for proving anything.
G-SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Special Right Triangles 30-60-90 and 45-45-90 triangle properties are not explicitly defined as part of the curriculum for Math II. However, using these properties in math III is extremely helpful when developing the Unit Circle. By learning these properties, students can save themselves time when solving problems involving these special right triangles. It will also allow students to more quickly develop the coordinates for the Unit Circle and to solve other application problems. This is also an ACT standard that is expected. Curriculum Plan Unit 1: Functions, Radicals and Exponents Domain & Range Function Characteristics Radical Operations Exponent Operations Solve Radical Equations -From a graph have students use both inequality and interval notion to describe domain and range. -Graph functions (linear, quadratic, exponential, square root, cube root, absolute value) and list characteristics of the function (domain, range, increasing, decreasing, relative max, relative min, positive negative, even, odd, end behavior, intercepts) -Simplify, add, subtract, mult, divide. -Laws of exponents -Change from radical to exponent -Include rational exponents -Include systems of linear and radical NC.M2.F-IF.4 NC.M2.F-IF.7 NC.M2.F-IF.9 NC.M2.F-IF.4 NC.M2.F-IF.7 NC.M2.F-IF.9 NC.M2.A-SSE.1 NC.M2.N-RN.3 NC.M2.N-RN.1 NC.M2.N-RN.2 NC.M2.A-REI.2 1 NC.M2.A-CED.3 NC.M2.F-BF.1 Inverse Variation -Include operations NC.M2.A-REI.2 1 NC.M2.A-SSE.1 NC.M2.A-CED.3 NC.M2.F-BF.1
Unit 2: Transformations Transformation Vocabulary Translations & Dilations -Mapping onto itself -Isometry -Discuss similarity and congruence -Discuss rigid motions for transformations NC.M2.G-CO.2 NC.M2.G-CO.3 NC.M2.G-CO.4 NC.M2.G-SRT.1b NC.M2.G-SRT.1c NC.M2.G-SRT.1d Rotations & Reflections -Discuss rigid motions NC.M2.G-CO.2 NC.M2.G-CO.3 NC.M2.G-CO.4 Transformation Combinations Dilating Linear Functions Function Transformation -Focus on combinations that perform mapping to original figure. -A linear function that goes through the origin will map to the same place. -A linear function that goes through a different y-intercept will be parallel. -Linear, Quadratic, Radical, Absolute Value, Exponential -Vertical & horizontal translations -Vertical stretches and compressions NC.M2.F-IF.2a NC.M2.F-IF.2b NC.M2.G-SRT.1a NC.M2.F-BF.3 NC.M2.F-IF.2a NC.M2.F-IF.2b Unit 3: Similarity and Congruence Triangle Theorems Parallel Lines Cut by Transversal Triangle Congruence Theorems -Triangle Angle Sum -Midsegment Properties -Isosceles Triangles -Angle pair relationships -Start two column proofs -Vertical angles - Discover through rigid motion - SSS, SAS, ASA, AAS, HL - Continue two column proofs NC.M2.G-CO.10 NC.M2.G-SRT.4 NC.M2.G-CO.9 NC.M2.G-CO.6 NC.M2.G-CO.7 NC.M2.G-CO.8 NC.M2.G-CO.9 Triangle Similarity -Discover AA through dilation NC.M2.G-SRT.8 NC.M2.G-SRT.3
Unit 4: Right Triangles Pythagorean Theorem -Use for problem solving NC.M2.G-SRT.4 Special Right Triangles -45-45-90-30-60-90 Right Triangle Trigonometry -Use for problem solving -Discovery from similarity -Model and solve problems NC.M2.G-SRT.8 NC.M2.G-SRT.12 NC.M2.G-SRT.6 Unit 5: Quadratics Unit 5A: Real Solutions & Graphs Polynomial Operation -Add, Subtract, Multiply NC.M2.A-APR.1 Factoring Solving Graphing Writing Applications - GCF - Quadratics - Difference of squares - Grouping -Methods of square root, factoring, quadratic formula -Real solutions only -Non-real solutions should be listed as such until 5B -Characteristics add axis of symmetry and vertex -Transformations connection to vertex form. -Talk about second differences in tables. -Convert from vertex to standard -Convert between standard and intercept form. -Projectile motion -Optimization Not explicitly in the standards but should be covered. NC.M2.F-IF.4 NC.M2.F-IF.7 NC.M2.F-BF.1 Unit 5B: Non-Real Solutions Imaginary numbers -represent imaginary and complex numbers -Add, subtract, mult (Honors) -Factoring (Honors) NC.M2.N-CN.1
Solving Writing Systems Unit 6: Probability Theoretical vs. Actual Probability Independent and Dependent Probability Tree Diagrams, Two-Way Tables Venn Diagrams Conditional Probability -Methods of square root, factoring, quadratic formula, and completing the square. -Include real and non-real answers -Convert between standard, vertex and intercept form -Solve systems of linear and quadratics -Run simulations to discover the more trials you run the more accurate theoretical probability gets. -Understand the difference between the two -Find through multiplication -Focus on skills of using tree diagram and two way tables to find probabilities -Students start making decisions of when to use tree diagrams versus two-way tables -Unions, intersections, and complements. -Must use and understand intersection (A B) and union (A U B) notation. -Construct and use Venn diagrams to describe probability -Tie instruction to previous instructional tools NC.M2.A-REI.4 NC.M2.A-SSE.3b NC.M2.A-SSE.3b NC.M2.F-IF.8 NC.M2.A-CED.3 NC.M2.A-REI.4 NC.M2.A-REI.7 NC.M2.S-IC.2 NC.M2.S-CP.3b NC.M2.S-CP.5 NC.M2.S-CP.8 NC.M2.S-CP.1 NC.M2.S-CP.3a NC.M2.S-CP.4 NC.M2.S-CP.1 NC.M2.S-CP.7 NC.M2.S-CP.1 NC.M2.S-CP.3b NC.M2.S-CP.5 NC.M2.S-CP.6 NC.M2.S-CP.7
Instructional Materials and Methods The honors course is taught based on the needs and abilities of the students. Units are taught with a variety of methods including whole class instruction, group activities, individual assignments, and presentations. The extensions listed above are things that have been done in previous honors courses. Each class is different and has different strengths and weaknesses. Each class may have different extensions. There also may be activities in this course that do not relate to the standards listed above. With the extensions in this honors course, students will complete activities both individually and in groups or partners that help them develop the necessary knowledge for standards in future courses. In developing these extensions, vertical alignment with Math III, Pre-Calculus, and AP Calculus standards were used as well as ACT standards. Students are expected to work independently and to have a drive to learn. Students should be driven and should want to learn for the sake of learning. Students should push themselves and come to class prepared. Assessment For Honors Math II, assessments have several different types of questions. Multiple choice, true or false, as well as short answer are several commonly used questions on the assessments. Test corrections are not used on a regular basis. Students are expected to know the material before the assessment. Tests for an honors course are more rigorous than in a standard course because they will be assessed on the extensions as well as the regular curriculum and standards. The final exam in this course is a North Carolina Final Exam. This is a state test. It is not created by the teacher. It counts as 25% of the students' final grade in the course, with each 9 week grade counting as 37.5%. There are released exam questions that will be used during exam review at the end of the course as well as throughout the course when the content is taught.