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# e-book Algebra II

Please hand in your solutions by the following Friday at in your assistant's box in HG J They will usually be corrected and returned in the following exercise class or, if not collected, returned to the box in HG J Please enroll in one of the exercise classes via echo.

For general information on the course, check the Course Catalogue. Algebra II Spring Lecturer Prof. Rahul Pandharipande. Coordinator Jennifer-Jayne Jakob. Exercise classes Wed Tue for students attending the electrodynamics course rooms: see below. The automorphism group of a field extension, solution of the cubic Rotman: A-1, pp , A II, Section F, pp Characters, fixed fields, solution of the quartic, Rotman: A-1, pp , A Fixed fields, Galois extensions, Rotman: A II, Sections G-H, pp Characterizations of Galois extensions, Normal field extensions, Rotman: A Fundamental Theorem of Galois theory Rotman: A II, Section H, pp Solvability by radicals Rotman: A 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. Students are asked to examine several variable expressions, interpret their meaning, and describe what quantities they each represent in the given context.

This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems. Students will recognize that dividing polynomials is similar to simplifying fractions. Students must use given means and standard deviations to approximate population percentages.

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Users are given the ability to define and change the coefficients and constants in order to observe resulting changes in the graph s. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. The task is to determine the relevant composite functions, their graphs, and the domain and range of each. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.

Key features of quadratic functions such as maximum values and zeros can often reveal important qualities of these phenomena. By the end of this tutorial, you should be able to find the zeros of a quadratic function and interpret their meaning in real-world contexts. Students interpret the composite function and determine values simply by using the tables of values. Depending on what aspect of the context we need to investigate, one expression of the quantity may be more useful than another.

This task contrasts the usefulness of four equivalent expressions. Students first have to confirm that the given expressions for the radioactive substance are equivalent. Then they have to explain the significance of each expression in the context of the situation. Fractals are characterized by self-similarity, smaller sections that resemble the larger figure.

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. 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.

Immediate feedback is provided, and for incorrect responses, each step of the solution is thoroughly modeled. 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.

Students need to be familiar with intercepts, and need to know what the vertex is. In particular, note that the purpose of the task is to have students generate the constraint equations for each part though the problem statements avoid using this particular terminology , and not to have students solve said equations.

If desired, instructors could also use this task to touch on such solutions by finding and interpreting solutions to the system of equations created in parts a and b.

Students are given a scenario and asked to determine the number of people required to complete the amount of work in the time described. An algebraic solution is possible but complicated; a numerical solution is both simpler and more sophisticated, requiring skilled use of units and quantitative reasoning.

By asking students to reason about solutions without explicitly solving them, we get at the heart of understanding what an equation is and what it means for a number to be a solution to an equation. The equations are intentionally very simple; the point of the task is not to test technique in solving equations, but to encourage students to reason about them. Students also have to pay attention to the scale on the vertical axis to find the correct match. The first and third graphs look very similar at first glance, but the function values are very different since the scales on the vertical axes are very different.

The task could also be used to generate a group discussion on interpreting functions given by graphs. Let's substitute the variable with a value Celsius temp to get the degrees in Fahrenheit.

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## Algebra II with Stotts

Great problem to practice with us! This video gives a great description of inconsistent, dependent, and independent systems. A consistent independent system of equations will have one solution. A consistent dependent system of equations will have infinite number of solutions, and an inconsistent system of equations will have no solution. This tutorial also provides information on how to distinguish a given system of linear equations as inconsistent, independent, or dependent system by looking at the slope and intercept. However, the context is not explicitly considered here.

This allows exploration of probabilities of multiple events as well as probability with and without replacement. The tabs above the applet provide access to 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. From this tutorial, students will learn the rules of imaginary numbers. Click below to open part 1. 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.

Being able to explain and justify your steps shows that you have mastered solving simple equations. Think about this process like walking on stones to cross a river, giving a reason for choosing each stone on which you place your foot. In this task, students may choose a representation that suits them and then reason from within that representation.

The purpose of this task is to give students practice in reading, analyzing, and constructing algebraic expressions, attending to the relationship between the form of an expression and the context from which it arises.

The context here is intentionally thin; the point is not to provide a practical application to kitchen floors, but to give a framework that imbues the expressions with an external meaning. While the context presents a classic example of exponential growth, it approaches it from a non-standard point of view.

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The goal of this task is to have students appreciate how different constants influence the shape of a graph. The model gives a surprisingly accurate estimate and this should be contrasted with linear and exponential models. Learn how light refraction and exponential growth help make candy colors just right! As a prerequisite to this lesson, students would need two years of high school algebra comfort with single variable equations and motivation to learn basic complex arithmetic.

## Algebra II Module 1 | EngageNY

Zager has included a complete introductory tutorial on complex arithmetic with homework assignments downloadable here. Also downloadable are some supplemental challenge problems. During the in-class portions of this interactive lesson, students will brainstorm on the outcome of the chaos game and practice calculating trajectories of difference equations.

In this example, students determine and then compare expressions that correspond to concentrations of various mixtures. Ultimately, students generalize the problem and verify conclusions using algebraic rather than numerical expressions. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

The model is realistic and provides a good context for students to practice work with exponential equations. This activity allows students to explore the effect of changing the sample size in an experiment and the effect of changing the standard deviation of a normal distribution.

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Tabs at the top of the page provide access to 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. 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. The function is linear and if simply looked at from a formulaic point of view, students might find the formula for the line and say that the domain and range are all real numbers.

Two of many methods for solving the system are presented. The first takes the given information to make three equations in three unknowns which can then be solved via algebraic manipulation to find the three numbers.