Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.
In this module, students synthesize and generalize what they have learned about a variety of function families. They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4). They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions. They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3). Students identify appropriate types of functions to model a situation. They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as, the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions, is at the heart of this module. In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.
In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.
In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.
Students will exploring how chaning the equation of a parabola on Desmos.com will give you similar graphs. These equations are all in vertex form and will ask students to determine the vertex of each equation. Students will all be asked to graph each of the equations in the families as well. Original worksheet created by Curt Sauer.
This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: The table below shows historical estimates for the population of London. Year18011821 18411861 18811901 1921 1939 1961 London population 1,100,000 1,60...
This lesson unit is intended to help teachers assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help teachers identify and help students who have the following difficulties: understanding how the factored form of the function can identify a graphŐs roots; understanding how the completed square form of the function can identify a graphŐs maximum or minimum point; and understanding how the standard form of the function can identify a graphŐs intercept.
This lesson unit is intended to help teachers assess how well students are able to: articulate verbally the relationships between variables arising in everyday contexts; translate between everyday situations and sketch graphs of relationships between variables; interpret algebraic functions in terms of the contexts in which they arise; and reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
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.
The goal of this task is to get students to focus on the shape of the graph of the equation y=ex and how this changes depending on the sign of the exponent and on whether the exponential is in the numerator or denominator. It is also intended to develop familiarity, in the case of f and k, with the functions which are used in logistic growth models, further examined in ``Logistic Growth Model, Explicit Case'' and ``Logistic Growth Model, Abstract Verson.''
This lesson is intended to help you assess how well students are able to:
Articulate verbally the relationships between variables arising in everyday contexts.Translate between everyday situations and sketch graphs of relationships between variables.Interpret algebraic functions in terms of the contexts in which they arise.Reflect on the domains of everyday functions and in particular whether they should be discrete or continuous.
The lesson includes a pre-assessment activity. The lesson is designed for students to work in small groups on a collaborative task, matching situations, sketch graphs, and algebraic functions. They refine the graphs and interpret the formulas to answer questions. Students then discuss as a whole-class what has been learned and the strategies used. Finally, a follow-up post-assessment is included and similar to the pre-assessment to use what they have learned.
This interactive simulation models the motion of a simple pendulum. Users can explore how pendulum motion is affected by changing length of the string, initial angle, and mass of the bob. Adjust the acceleration due to gravity to simulate pendulum motion on another planet. Energy bar graphs can be displayed in stepped motion alongside the swinging pendulum to get a clear picture of kinetic/potential energy conversion. Click on "Forces" to see free body diagrams. Advanced learners can view graphs of angular position, angular velocity, and angular acceleration as well. The model is simple enough for middle school students to manipulate, yet also provides an array of robust tools that render it appropriate for introductory physics courses. See Related Materials for a multi-day module on simple harmonic motion (Science NetLinks) and for instructions on installing and running the cost-free EJS Modeling and Authoring Tool. This applet was created with EJS, Easy Java Simulations, a modeling tool that allows users without formal programming experience to generate computer models and simulations.
This interactive simulation offers a way for students to explore the connection between uniform circular motion and simple harmonic motion. The display shows two blocks on springs oscillating horizontally, and two balls traveling in uniform motion in a circular path. The user sets initial values for the blocks: amplitude, mass, and spring constant. The two balls are automatically set to the same values. Students are able to see that the circular motion of each ball corresponds to the motion of the blocks, thus promoting understanding of the basic equation for objects undergoing simple harmonic motion. To extend the learning, users can set values for the phase angles of each block. Also included by the author is a set of suggested activities to accompany the simulation. See Related Materials for an extensive online multimedia tutorial from PhysClips on the topic of simple harmonic motion. This applet was created with EJS, Easy Java Simulations, a modeling tool that allows users without formal programming experience to generate computer models and simulations.