# Mathematical Analysis Honors   (#1201300)

## General Course Information and Notes

### General Notes

Honors and Advanced Level Course Note: Advanced courses require a greater demand on students through increased academic rigor.  Academic rigor is obtained through the application, analysis, evaluation, and creation of complex ideas that are often abstract and multi-faceted.  Students are challenged to think and collaborate critically on the content they are learning. Honors level rigor will be achieved by increasing text complexity through text selection, focus on high-level qualitative measures, and complexity of task. Instruction will be structured to give students a deeper understanding of conceptual themes and organization within and across disciplines. Academic rigor is more than simply assigning to students a greater quantity of work.

English Language Development ELD Standards Special Notes Section:
Teachers are required to provide listening, speaking, reading and writing instruction that allows English language learners (ELL) to communicate information, ideas and concepts for academic success in the content area of Mathematics. For the given level of English language proficiency and with visual, graphic, or interactive support, students will interact with grade level words, expressions, sentences and discourse to process or produce language necessary for academic success. The ELD standard should specify a relevant content area concept or topic of study chosen by curriculum developers and teachers which maximizes an ELL’s need for communication and social skills. To access an ELL supporting document which delineates performance definitions and descriptors, please click on the following link:

### General Information

Course Number: 1201300
Abbreviated Title: MATH ANALYSIS HON
Number of Credits: Half credit (.5)
Course Length: Semester (S)
Course Attributes:
• Honors
Course Level: 3
Course Status: Course Approved

## Educator Certifications

One of these educator certification options is required to teach this course.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this course.

## Original Student Tutorials

Dilations...The Effect of k on a Graph:

Visualize the effect of using a value of k in both kf(x) or f(kx) when k is greater than zero in this interactive tutorial.

Type: Original Student Tutorial

Reflections...The Effect of k on a Graph:

Learn how reflections of a function are created and tied to the value of k in the mapping of f(x) to -1f(x) in this interactive tutorial.

Type: Original Student Tutorial

Translations...The Effect of k on the Graph:

Explore translations of functions on a graph that are caused by k in this interactive tutorial. GeoGebra and interactive practice items are used to investigate linear, quadratic, and exponential functions and their graphs, and the effect of a translation on a table of values.

Type: Original Student Tutorial

Long Division With Polynomials:

Use long division to rewrite a rational expression of the form a(x) divided by b(x) in the form q(x) plus the quantity r(x) divided by b(x), where a(x), b(x), q(x), and r(x) are polynomials.

Type: Original Student Tutorial

The graph of a quadratic equation is called a parabola [puh-ra-bow-luh]. The key features we will focus on in this tutorial are the vertex (a maximum or minimum extreme) and the direction of its opening. You will learn how to examine a quadratic equation written in vertex form in order to distinguish each of these key features.

Type: Original Student Tutorial

## Perspectives Video: Experts

Type: Perspectives Video: Expert

Mathematically Exploring the Wakulla Caves:

The tide is high!  How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a fresh water spring 20 miles from each other?

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Hacking the Matrices (in your Kinect) for Real-Time 3D Scanning:

What's the point to learning about matrices? You can hack gaming devices for off-the-shelf real time 3D visualization!

Type: Perspectives Video: Professional/Enthusiast

Making Color with Matrices:

Did you know that computers use matrices to represent color? Learn how computer graphics work in this video.

Type: Perspectives Video: Professional/Enthusiast

## Presentation/Slideshow

Matrices:

This resource is a PowerPoint presentation and a form for guided note taking to be used while viewing the presentation about Matrix Operations. It begins by defining matrices and identifying types of matrices. It then goes into how to add, subtract, and multiply matrices, including how to use scalar multiplication. The final portion deals with finding the determinants of 2x2 and 3x3 matrices and Inverse Matrices.

Type: Presentation/Slideshow

The task is intended to address sample space, independence, probability distributions and permutations/combinations.

Alex, Mel, and Chelsea Play a Game:

This task combines the concept of independent events with computational tools for counting combinations, requiring fluent understanding of probability in a series of independent events.

Random Walk III:

The task provides a context to calculate discrete probabilities and represent them on a bar graph.

Random Walk IV:

This problem solving task gives a situation where the numbers are too large to calculate, so abstract reasoning is required in order to compare the different probabilities.

Finding Parabolas through Two Points:

This problem-solving task challenges students to find all quadratic functions described by given equation and coordinates, and describe how the graphs of those functions are related to one another.

Temperatures in Degrees Fahrenheit and Celsius:

Temperature conversions provide a rich source of linear functions which are encountered not only in science but also in our every day lives when we travel abroad. The first part of this task provides an opportunity to construct a linear function given two input-output pairs. The second part investigates the inverse of a linear function while the third part requires reasoning about quantities and/or solving a linear equation.

Trina's Triangles:

In this task students investigate and ultimately prove the validity of the method of generating Pythagorean Triples that involves the polynomial identity (x2+y2)2=(x2−y2)2+(2xy)2.

Zeroes and factorization of a general polynomial:

In this task, students are asked to show or verify four theorems related to roots, zeroes, and factors of polynomial functions. The Fundamental theorem of Arithmetic is also mentioned. This task builds on "Zeroes and factorization of a quadratic function'' parts I and II.

Compounding with a 100% Interest Rate:

This task provides an approximation, and definition, of e, in the context of more and more frequent compounding of interest in a bank account. The approach is computational.

Zeroes and factorization of a non polynomial function:

For a polynomial function f, if f(0)=0 then the polynomial f(x) is divisible by x. This fact is shown and then generalized in "Zeroes of a quadratic polynomial I, II" and "Zeroes of a general polynomial.'' Here, divisibility tells us that the quotient f(x)/x will still be a nice function -- indeed, another polynomial, save for the missing point at x=0. The goal of this task is to show via a concrete example that this nice property of polynomials is not shared by all functions. The non-polynomial function F given by F(x)=|x| is a familiar function for which property does not hold: even though F(0)=0, the quotient F(x)/x behaves badly near x=0. Indeed, its graph is broken into two parts which do not connect at x=0.

Building a quadratic function from f(x) = x^2:

This task aims for students to understand the quadratic formula in a geometric way in terms of the graph of a quadratic function.

Building an explicit quadratic function by composition:

This task is intended for instruction and to motivate "Building a general quadratic function." This task assumes that the students are familiar with the process of completing the square.

A Sum of Functions:

In this example, students are given the graph of two functions and are asked to sketch the graph of the function that is their sum. The intent is that students develop a conceptual understanding of function addition.

Graphs of Compositions:

This task addresses an important issue about inverse functions. In this case the function f is the inverse of the function g but g is not the inverse of f unless the domain of f is restricted.

Zeroes and factorization of a quadratic polynomial II:

This task continues "Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in "Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. In the quadratic case, an alternative argument for why there can be at most two roots can be given using the quadratic formula and this is done in the second solution below.
This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions. Students who are familiar with the quadratic formula should be encouraged to think about the first solution which extends to polynomials of higher degree where formulas for the roots are either very complex or not possible to find.

Crude Oil and Gas Mileage:

This task asks students to write expressions for various problems involving distance per units of volume.

Zeroes and factorization of a quadratic polynomial I:

For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x-r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact. The teacher should pay close attention to the logic used in the solution to part (c) where the divisibility of ax2+bx+c by x-r is obtained not by performing long division but by using the result of long division of these polynomials; namely, that said division will result in an expression of the following form:
ax2+bx+c=(x-r)l(x)+k
where l is a linear polynomial and k is a number.
This task could be used either for assessment or for instructional purposes. If it is used for assessment, parts (a) and (b) are more suitable than part (c). Each of the questions in this task could be formulated as an if and only if statement but the other implication, namely that f(x) is divisible by x-r if and only if r is a root of f. The direction not presented in this task is more straightforward and so has been left out.

Flu on Campus:

The context of this example is the spread of a flu virus on campus and the related sale of tissue boxes sold. Students interpret the composite function and determine values simply by using the tables of values.

Combined Fuel Efficiency:

In this example, fuel efficiency of a car can be analyzed by using rational expressions and operations with rational expressions.

In this resource, a method of deriving the quadratic formula from a theoretical standpoint is demonstrated. This task is for instructional purposes only and builds on "Building an explicit quadratic function."

Compounding with a 5% Interest Rate:

This task develops reasoning behind the general formula for balances under continuously compounded interest. While this task itself specifically address the standard (F-BF), building functions from a context, an auxiliary purpose is to introduce and motivate the number e, which plays a significant role in the (F-LE) domain of tasks.

Students compare graphs of different quadratic functions, then produce equations of their own to satisfy given conditions.

This exploration can be done in class near the beginning of a unit on graphing parabolas. Students need to be familiar with intercepts, and need to know what the vertex is. It is effective after students have graphed parabolas in vertex form (y=a(x–h)2+k), but have not yet explored graphing other forms.

Graphs of Power Functions:

This task requires students to recognize the graphs of different (positive) powers of x.

Transforming the Graph of a Function:

This problem solving task examines, in a graphical setting, the impact of adding a scalar, multiplying by a scalar, and making a linear substitution of variables on the graph of the function f. This resource also includes standards alignment commentary and annotated solutions.

Temperature Conversions:

Unit conversion problems provide a rich source of examples both for composition of functions (when several successive conversions are required) and inverses (units can always be converted in either of two directions).

Susita's Account:

This task asks students to determine a recursive process from a context. Students who study computer programming will make regular use of recursive processes.

Summer Intern:

This task asks students to use proportions of mass and volume to create ideal brine for saltwater fish tanks. It also asks students to compare graphs.

Skeleton Tower:

This problem is a quadratic function example. The other tasks in this set illustrate MAFS.912.F.BF.1.1.a in the context of linear, exponential, and rational functions.

Medieval Archer:

The task addresses the first part of standard MAFS.912.F-BF.2.3: "Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative)."

Lake Algae:

The purpose of this task is to introduce students to exponential growth. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.

The Canoe Trip, Variation 2:

The primary purpose of this task is to lead students to a numerical and graphical understanding of the behavior of a rational function near a vertical asymptote, in terms of the expression defining the function.

The Canoe Trip, Variation 1:

The purpose of this task is to give students practice constructing functions that represent a quantity of interest in a context, and then interpreting features of the function in the light of the context. It can be used as either an assessment or a teaching task.

Identifying Even and Odd Functions:

This task asks students to determine whether a the set of given functions is odd, even, or neither.

The Missing Coefficient:

The purpose of this task is to emphasize the use of the Remainder Theorem (a discussion of which should obviously be considered as a prerequisite for the task) as a method for determining structure in polynomial in equations, and in this particular instance, as a replacement for division of polynomials.

## Tutorials

Graphs and Solutions of Functions in Quadratic Equations:

You will learn how the parent function for a quadratic function is affected when f(x) = x2.

Type: Tutorial

Binomial Theorem:

This video tutorial gives an introduction to the binomial theorem and explains how to use this theorem to expand binomial expressions.

Type: Tutorial

Pascal's Triangle for Binomial Expansion:

This tutorial shows students how to use Pascal's triangle for binomial expansion.

Type: Tutorial

Graphs of second degree polynomials:

In this tutorial, students will look at input and output values of quadratic functions to help them understand why the graph of a second degree polynomial curves.

Type: Tutorial

Division of Polynomials:

This resource discusses dividing a polynomial by a monomial and also dividing a polynomial by a polynomial using long division.

Type: Tutorial

Geometric sequence or progression:

We will learn how to write a geometric sequence.

Type: Tutorial

Geometric series:

Geometric series

Type: Tutorial

Finding the nth term in a recursively defined sequence:

Finding the 5th term in recursively defined sequence

Type: Tutorial

Graphing Quadractic Functions in Vertex Form:

This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.

Type: Tutorial

This tutorial will help the learners to graph the equation of the quadratic function using the coordinates of the vertex of a parabola adn its x- intercepts.

Type: Tutorial

Graphing Exponential Equations:

This tutorial will help you to learn about the exponential functions by graphing various equations representing exponential growth and decay.

Type: Tutorial

Compound Probability with Dependent Events:

This video describes using multiplication to find the compound probability of dependent events.

Type: Tutorial

Dividing Polynomials:

This tutorial will help the learners practice division of polynomials. Students will recognize that dividing polynomials is similar to simplifying fractions.

Type: Tutorial

Refraction of Light:

This resource explores the electromagnetic spectrum and waves by allowing the learner to observe the refraction of light as it passes from one medium to another, study the relation between refraction of light and the refractive index of the medium, select from a list of materials with different refractive indicecs, and change the light beam from white to monochromatic and observe the difference.

Type: Tutorial

Human Eye Accommodation:

• Observe how the eye's muscles change the shape of the lens in accordance with the distance to the object being viewed
• Indicate the parts of the eye that are responsible for vision
• View how images are formed in the eye

Type: Tutorial

Concave Spherical Mirrors:

• Learn how a concave spherical mirror generates an image
• Observe how the size and position of the image changes with the object distance from the mirror
• Learn the difference between a real image and a virtual image
• Learn some applications of concave mirrors

Type: Tutorial

Convex Spherical Mirrors:

• Learn how a convex mirror forms the image of an object
• Understand why convex mirrors form small virtual images
• Observe the change in size and position of the image with the change in object's distance from the mirror
• Learn some practical applications of convex mirrors

Type: Tutorial

Color Temperature in a Virtual Radiator:

• Observe the change of color of a black body radiator upon changes in temperature
• Understand that at 0 Kelvin or Absolute Zero there is no molecular motion

Type: Tutorial

Solar Cell Operation:

This resource explains how a solar cell converts light energy into electrical energy. The user will also learn about the different components of the solar cell and observe the relationship between photon intensity and the amount of electrical energy produced.

Type: Tutorial

Electromagnetic Wave Propagation:

• Observe that light is composed of oscillating electric and magnetic waves
• Explore the propagation of an electromagnetic wave through its electric and magnetic field vectors
• Observe the difference in propagation of light of different wavelengths

Type: Tutorial

Basic Electromagnetic Wave Properties:

• Explore the relationship between wavelength, frequency, amplitude and energy of an electromagnetic wave
• Compare the characteristics of waves of different wavelengths

Type: Tutorial

Geometrical Construction of Ray Diagrams:

• Learn to trace the path of propagating light waves using geometrical optics
• Observe the effect of changing parameters such as focal length, object dimensions and position on image properties
• Learn the equations used in determining the size and locations of images formed by thin lenses

Type: Tutorial

## Video/Audio/Animations

Will an Ice Cube Melt Faster in Freshwater or Saltwater?:

With an often unexpected outcome from a simple experiment, students can discover the factors that cause and influence thermohaline circulation in our oceans. In two 45-minute class periods, students complete activities where they observe the melting of ice cubes in saltwater and freshwater, using basic materials: clear plastic cups, ice cubes, water, salt, food coloring, and thermometers. There are no prerequisites for this lesson but it is helpful if students are familiar with the concepts of density and buoyancy as well as the salinity of seawater. It is also helpful if students understand that dissolving salt in water will lower the freezing point of water. There are additional follow up investigations that help students appreciate and understand the importance of the ocean's influence on Earth's climate.

Type: Video/Audio/Animation

Basic Linear Function:

This video demonstrates writing a function that represents a real-life scenario.

Type: Video/Audio/Animation

Graphing Lines 1:

Khan Academy video tutorial on graphing linear equations: "Algebra: Graphing Lines 1"

Type: Video/Audio/Animation

## Virtual Manipulatives

Slope Slider:

In this activity, students adjust slider bars which adjust the coefficients and constants of a linear function and examine how their changes affect the graph. The equation of the line can be in slope-intercept form or standard form. This activity allows students to explore linear equations, slopes, and y-intercepts and their visual representation on a graph. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Graphing Equations Using Intercepts:

This resource provides linear functions in standard form and asks the user to graph it using intercepts on an interactive graph below the problem. Immediate feedback is provided, and for incorrect responses, each step of the solution is thoroughly modeled.

Type: Virtual Manipulative

Graphing Lines:

Allows students access to a Cartesian Coordinate System where linear equations can be graphed and details of the line and the slope can be observed.

Type: Virtual Manipulative

Data Flyer:

Using this virtual manipulative, students are able to graph a function and a set of ordered pairs on the same coordinate plane. The constants, coefficients, and exponents can be adjusted using slider bars, so the student can explore the affect on the graph as the function parameters are changed. Students can also examine the deviation of the data from the function. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Function Matching:

This is a graphing tool/activity for students to deepen their understanding of polynomial functions and their corresponding graphs. This tool is to be used in conjunction with a full lesson on graphing polynomial functions; it can be used either before an in depth lesson to prompt students to make inferences and connections between the coefficients in polynomial functions and their corresponding graphs, or as a practice tool after a lesson in graphing the polynomial functions.

Type: Virtual Manipulative

Function Flyer:

In this online tool, students input a function to create a graph where the constants, coefficients, and exponents can be adjusted by slider bars. This tool allows students to explore graphs of functions and how adjusting the numbers in the function affect the graph. Using tabs at the top of the page you can also access supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Number Cruncher:

In this activity, students enter inputs into a function machine. Then, by examining the outputs, they must determine what function the machine is performing. This activity allows students to explore functions and what inputs are most useful for determining the function rule. This activity includes supplemental materials, including background information about the topics covered, a description of how to use the application, and exploration questions for use with the java applet.

Type: Virtual Manipulative

Curve Fitting:

With a mouse, students will drag data points (with their error bars) and watch the best-fit polynomial curve form instantly. Students can choose the type of fit: linear, quadratic, cubic, or quartic. Best fit or adjustable fit can be displayed.

Type: Virtual Manipulative

Equation Grapher:

This interactive simulation investigates graphing linear and quadratic equations. Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph(s).

Type: Virtual Manipulative

Geometric and Harmonic Series- Limits:

This applet allows users to set up various geometric series with a visual representation of the successive terms, and the corresponding sum of those terms.

Type: Virtual Manipulative

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this course.