# Honors Level Course Overview - Math III

• On this page you will find resources that will help you understand the difference between a standard and an honors Math III course.  You will also find resources that will guide you throughout the semester in understanding the curriculum, instruction, and assessments.

Curriculum Content

Honors Math III progresses from the standards learned in Math I and Math II. In addition to these standards, Math III extends to include algebraic concepts such as functions and their inverses, an introduction to logarithmic functions and their relationship to exponential functions, additional operations with polynomials, rational functions, and trigonometric functions and the unit circle. Math III also inclues the geometric concepts of circles and parallelograms.

This course will allow students to develop skills with functions and to recognize additional properties of those graphs beyond what was covered in Math II. Students will discover through exploratory activities, appropriate use of technology, group work and hands on activities.

Honors Math III will have more rigorous standards than a standard Math III course, and the expectation will be that students will learn the basic standards quickly and will be ready to dig deeper into the content, allowing time for various extensions.

Students taking Honors Math III are most likely the students who will take Honors PreCalculus and either AP Calculus or AP Statistics. This course will focus on skills needed for Honors PreCalculus and AP Calculus, and will ensure students are prepared for the rigors of those courses. 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 the standards of Honors PreCalculus and AP Calculus, as well as ACT and SAT standards.

Standards and Objectives

Concavity -

NC.M3.F.IF.7 - Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases including: domain and range; intercepts; intervals where the function is increasing , decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period and discontinuities.

Concavity is a characteristic of a graph that features heavily when studying the applications of differential calculus. In AP Calculus, students will be asked to analyze a graph based on it’s concavity, and to draw a graph given various characteristics including concavity. In this course the basic concept of concavity will be introduced, and in Honors PreCalculus it will be explored further.

Orthocenter –

NC.M3.G-CO.10 - Verify experimentally properties of the centers of triangles (centroid, incenter, and circumcenter).

Like the centroid, incenter and circumcenter, the orthocenter is a center of a triangle. It is created at the intersection of three altitudes. Honors students will discover the unique properties of the orthocenter via a self guided activity and will compare these properties to the properties of the other centers.

Factoring perfect cubes –

NC.M3.A-SSE.2 - Use the structure of an expression to identify ways to write equivalent expressions. For example, see x4 – y4 as (x2) 2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Factoring perfect cubes is a topic that comes up in Calculus during the study of limits. Limits begins the study of differential calculus. Often a limit cannot be directly evaluated at a specific point because of a hole at that point. Factoring the numerator or denominator of a rational function can revel the hole. Sometimes this factoring requires the factoring of perfect cubes.

Add, subtract and multiply complex numbers -

Operations with complex numbers were moved to Math IV beginning in the 2016-2017 school year. However, these operations are an extension of polynomial operations and allow students to discuss complex roots and the effects of those roots on the graph. For Honors Math III, these operations will be discussed.

Writing/building polynomials given complex and radical zeros –

NC.M3.A-APR.3 - Understand the relationship among factors of a polynomial expression, the solutions of a polynomial equation and the zeros of a polynomial function.

NC.M3.A-CED.3 - Create systems of equations and/or inequalities to model situations in context

It is within the scope of Math III to build polynomial functions given real, rational zeros. The honors course with extend this concept to include complex and radical zeros. The purpose of this is to build the student’s algebra skills so that they are comfortable with the algebra requirements of calculus.

Use synthetic division to uncover complex solutions –

NC.M3.A-APR.3 - Understand the relationship among factors of a polynomial expression, the solutions of a polynomial equation and the zeros of a polynomial function.

NC.M3.A-REI.1 - Justify a solution method for equations and explain each step of the solving process using mathematical reasoning

This extension is done to reinforce skills like the quadratic formula and completing the square learned in Math I and Math II so that students remember these skills. Not only do these skills show up on the Math III NCFE, they are important in the algebraic manipulation of calculus problems.

Consider rational denominators beyond linear –

NC.M3.A-APR.7 - Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.

1. Add and subtract two rational expressions, and , where the denominators of both and are linear expressions.
2. Multiply and divide two rational expressions.

By considering nonlinear denominators students can practice factoring skills, including factor perfect cubes (a Honors Math 3 extension discussed earlier). These skills are necessary in Calculus when solving limits algebraically. Nonlinear denominators of rational functions also occur when integrating by parts with partial fractions.

Simplify complex fractions -

NC.M3.A-APR.7 - Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.

1. Add and subtract two rational expressions, and , where the denominators of both and are linear expressions.
2. Multiply and divide two rational expressions.

Complex fractions also come up when evaluating limits algebraically in calculus. By discussing complex fractions in both Math III and again in PreCalculus, students are very familiar with them in calculus.

Discuss increasing/decreasing and concavity as it pertains to sine and cosine functions -

NC.M3.F-TF.2 - Build an understanding of trigonometric function by using tables, graphs and technology to represent the cosine and sine functions.

1. Interpret the sine functions as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y-coordinate.

NC.M3.F.IF.7 - Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases including: domain and range; intercepts; intervals where the function is increasing , decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period and discontinuities.

Students get the opportunity to discuss concavity again as it relates to the sine and cosine function. This is to facilitate the PreCalculus discussion of the tangent, cotangent, secant and cosecant graphs, all of which are related to sine and cosine. A solid understand of trigonomentry is required of students in Calculus. The foundation of this understanding is laid in Math II and Math III, and continued in PreCalculus.

Complete the square with leading coefficients other than 1 –

NC.M3.G-GPE.1 - Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Circles are the only conic sections left in Math III. All other conic sections have moved to PreCalculus as of the 2016-2017 school year. All conic sections require the student to complete the square to write the equation of the conic. However, parabolas, ellipses and hyperbolas often have leading coefficients other than one. By introducing this concept in Honors Math III, students have more time to master an important PreCalculus skill.

Curriculum Plan

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 discovery activities, appropriate technology and individual assignments. Each class is different, and has different strengths and weaknesses. Therefore, extensions may vary by class. 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 future math courses. In developing these extensions, vertical alignment with Honors PreCalculus and AP Calculus were used.

An example of a Honors Math III extension is the Spagetti Sine activity, whereby students discover for themselves the relationship between the unit circle and the sine curve. Students use raw spaghetti to measure the distance from the x axis to the point on the unit circle, and using the measurements a sine curve is formed.

Spagetti Sine

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 III, assessments will be a mix of tests, quizzes and short pop quizzes. Students are asked to complete several different types of questions like multiple choice and short answer. Test corrections and retests are not used. Students are expected to know the material before the assessment. Assessments 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. Alternative assessments that may be used include short “ticket out the door”, or individual or group projects assessed on a rubric.

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.