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: https://cpalmsmediaprod.blob.core.windows.net/uploads/docs/standards/eld/ma.pdf
Course Number: 1201315
Abbreviated Title: ANALYSIS OF FUNC HON
Number of Credits: Half credit (.5)
Course Length: Semester (S)
Course Type: Core Academic Course
Course Level: 3
Course Status: Terminated
Grade Level(s): 9,10,11,12
Graduation Requirement: Mathematics
One of these educator certification options is required to teach this course.
Learn how to evaluate a speaker's point of view, reasoning, and use of evidence. In this interactive tutorial, you'll examine Abraham Lincoln's "Gettysburg Address" and evaluate the effectiveness of his words by analyzing his use of reasoning and evidence.
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 with this interactive tutorial.
In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.
The purpose of this task is to give students experience modeling a real-world example of exponential growth, in a context that provides a vivid illustration of the power of exponential growth, for example the cost of inaction for a year. There is an opportunity for further discussion based on part (c), since the ratio of costs from one year to the next is the same in each part.
The coffee cooling experiment is a popular example of an exponential model with immediate appeal. The model is realistic and provides a good context for students to practice work with exponential equations.
This problem introduces the method used by scientists to date certain organic material. It is based not on the amount of the Carbon 14 isotope remaining in the sample but rather on the ratio of Carbon 14 to Carbon 12. This ratio decreases, hypothetically, at a constant exponential rate as soon as the organic material has ceased to absorb Carbon 14, that is, as soon as it dies.
The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology, among other places. Students should be guided to recognize the use of the natural logarithm when the exponential function has the given base of e, as in this problem. Note that the purpose of this task is algebraic in nature -- closely related tasks exist which approach similar problems from numerical or graphical stances.
This task provides a real world context for interpreting and solving exponential equations. There are two solutions provided for part (a). The first solution demonstrates how to deduce the conclusion by thinking in terms of the functions and their rates of change. The second approach illustrates a rigorous algebraic demonstration that the two populations can never be equal.
This is a simple task touching on two key points of functions. First, there is the idea that not all functions have real numbers as domain and range values. Second, the task addresses the issue of when a function admits an inverse, and the process of "restricting the domain" in order to achieve an invertible function.
The task addresses knowledge related to interpreting forms of functions derived by factoring or completing the square. It requires students to pay special attention to the information provided by the way the equation is represented as well as the sign of the leading coefficient, which is not written out explicitly, and then to connect this information to the important features of the graph.
The problem presents a context where a quadratic function arises. Careful analysis, including graphing of the function, is closely related to the context. The student will gain valuable experience applying the quadratic formula and the exercise also gives a possible implementation of completing the square.
The purpose of this task is to construct and use inverse functions to model a real-life context. Students choose a linear function to model the given data, and then use the inverse function to interpolate a data point.
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.
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.
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.
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.
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.
The purpose of this task is to help students see the "why" behind properties of logs that are familiar but often just memorized (and quickly forgotten or misremembered). The task focuses on the verbal definition of the log, helping students to concentrate on understanding that a logarithm is an exponent, as opposed to completing a more computational approach.
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.
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.
In this task, students explore the inverse relationship between an exponential function and a logarithmic function. The task is to determine the relevant composite functions, their graphs, and the domain and range of each.
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.
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).
In this task, students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible.
This task illustrates several components of standard MAFS.912.F-BF.2.4.c: Find Inverse Functions. Here, instead of presenting two functions and asking the students to decide which on is invertible, students are asked to complete a table of input-output pairs for the functions in such a way that one of the functions is invertible and the other one is not.
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 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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
Vetted resources caregivers can use to help students learn the concepts and skills in this course.
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