How To Teach Algebra Ii
Teaching a higher-level math course can be challenging, particularly for those of us who took very different versions of math when we were in high school. The good news is that you do not have to do this alone. There are many resources available, and there are homeschoolers who have walked the path already.
Here are some steps to get you started toward success with high school Algebra 2:
How Are Linear Equations Used In Everyday Life
Many students resent having to learn algebra in high school or college because they don’t see how it applies to real life. Yet, the concepts and skills of Algebra 2 provide invaluable tools for navigating business solutions, financial problems and even everyday dilemmas. The trick to successfully using Algebra 2 in real life is determining which situations call for which formulas and concepts. Luckily, the most common real life problems call for widely applicable and highly recognizable techniques.
Use linear equations to determine how much of something you can afford when a service involves both a rate and a flat fee. For instance, if you want to know how many months of a gym membership you can afford, write out an equation with the monthly fee times “X” number of months plus the amount the gym charges up front to join and set it equal to your budget. If the gym charges $25/month, there is a $75 flat fee, and you have a budget of $275, your equation would look like this: 25x + 75 = 275. Solving for x tells you that you can afford eight months at that gym.
How To Draw Y^2 = X
By Murray Bourne, 09 Apr 2009
Nuaja, a subscriber to the IntMath Newsletter, wrote recently:
How do I know how the graph should look like: For example: y2 = x – 2?
The first thing I recognize in that equation is the y2 term, which tells me it will be a parabola.
Let’s start with the most basic parabola y = x2 and build up to the required answer.
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What Does It Look Like
A line written in slope-intercept form takes this form:
Sometimes you’ll even hear people refer to slope-intercept as “y=mx+b form” because it’s so well known. Let’s examine what each term means. Remember that you are plotting this graph on an x-y plane, so the variables x and y represent geometric points on the graph. The constant ‘m’ is the slope of the graph — notice how it multiplies your x term. That’s because the slope of a graph determines how much ‘y’ changes as a function of ‘x’. The final term, b, is the y-intercept. That’s the point at which the line crosses the y-axis, where x=0. If you plug in 0 for x, you’ll see that all you have is y=b.
What Exactly Do Elements Of $f / $ Look Like
For the field $F$, integral domain $F$, some element $c$ which is not contained in $F$ and the minimal polynomial of $c$, $f$ I know that we have the isomorphism
$$F \cong F / )$$
where $)$ is the ideal generated by the polynomial $f$, but I am having a lot of difficulty in comprehending what elements of $F / )$ actually look like. Can someone provide a few concrete examples, or is there a set-theory type definition which describes all of these elements?
I also struggled with this when I was learning ring theory. So let’s start with the ideal
$$)=\$$this is just all elements in $F$ which are divisible by $f$.Now for the quotient, remember in a ring we have two operation $+, \cdot$. For quotient rings, we take the the ideal $)$ and move it around by addition of other elements in the ring. All elements that are divisible by $)$ are seen zero since the get absorbed by the ideal .
So what does an element in $F/)$ look like?Well take an $g\in F$ and consider$$g+)=\$$and that’s it.
Now, let’s try an example maybe in the context of what you’re talking about.We know of course that $\sqrt\not\in \mathbb$ but we know that $\sqrt$ is a root of the polynomial $x^2-2\in \mathbb$ and it is the minimal polynomial of this number. We can show that the field extension$$\mathbb \cong \mathbb/$$this is done by constructing the isomorphism and is pretty standard.
$$9x^+x^-3x+6\equiv9x^+x^-3x+6-9\left\equiv x^+6x+6\mod x^3-x$$
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Why Choose T4l Algebra Ii Homeschool Curriculum
Our Algebra II course is taught using a combination of multimedia lessons, instructional videos, quizzes, tests and both online and offline projects. Students can expect to see various concepts being covered including functions, equations, probability, statistics, matrices, and more.
Selecting Time4Learnings Algebra 2 online course opens possibilities for customizing learning to your childs strengths and needs. Thousands of families use our program as a complete full-year homeschooling curriculum, while others prefer it as an Algebra 2 summer course to help struggling students or to help them get ahead.
Below are just a few reasons why many parents and students prefer Time4Learnings Algebra II curriculum:
|As a Full Curriculum|
What To Do About Algebra 2
A recent Forbes article calls for replacing Algebra 2 with computational thinking. What’s good about that idea and what needs more careful thought?
This week in Forbes magazine, Tom VanderArk renewed an ongoing and often intense debate ;about the role of second-year algebra in math curricula in high schools and colleges. It’s far from the first time that people have questioned whether Algebra 2 should be required of all students in a school district or college setting. There was pushback in 2013 from the state of Florida and more recently in California. In fact right around 2012-2013, when the Common Core State Standards were emerging , resistance to “algebra 2 for all” also started emerging both in these specific cases and in more general op-eds, like this article in Slate and this one in the New York Times, and the issue doesn’t seem to be losing any steam.
The Forbes article in a nutshell says the following:
Before I get into the specifics, just a note: I met Tom VanderArk at the Steelcase Active Learning Symposium last fall, and I was really impressed by him. He’s a smart guy with interesting ideas about the role of technology in our lives, including education, and I learned a lot from his keynote address. I was also grateful he took some time out to talk with me about my sabbatical at Steelcase, and he was ;gracious and generous, despite dealing with the usual last-minute chaos of getting ready for a big talk.
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Basic Knowledge Of Polynomial Functions
A polynomial is a mathematical expression constructed with constants and variables using the four operations:
In other words, we have been calculating with various polynomials all along. When two polynomials are divided it is called a rational expression.
In such cases you must be careful that the denominator does not equal zero. Division by zero is not defined and thus x may not have a value that allows the denominator to become zero. Otherwise, any other value may be substituted for x.
x must not have the value of 6 since 6-6=0.
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Great for High School Math
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Unit : Exponents And Logarithms
Given a sequential search and a binary search, students will analyze the efficiency of each search and classify them as linear or logarithmic. Â Given a sampling of sort functions, students will analyze the efficiencies of each sort and classify them according to the graph of sample size vs how many swaps. Students will present their code analysis.
Loan Amortization Students create a program that will solve exponential and logarithmic equations demonstrating how interest on a loan is compounded over a long period of time using an appropriate data structure i.e. arrays.
Catch the Fever Write a program to model the spread of a virus. Based on the initial percent of the population who is immune, virulence , duration of infection , rate of transmission they will show the percent of people who are sick, immune and total population still alive .
“Research a programming code that utilizes recursion as a programming technique to solve complex problems. Compare the recursion with problem solving techniques you already know.” Students will read and respond to 3 peer blog entries.
Skill Practice sets from State Approved Algebra 2 Textbook
Unit : Rational Expressions And Equations
Programming Task 1
The Unit 3 Programming Task Key Assignment factored polynomials of degree 2 or 3. Building on the implementation of that key assignment and the additional Algebra 2 knowledge of simplifying rational expressions, students will write and run a program that will: Â 1). demonstrate knowledge of the use of various operator, operator precedence, data types, variables, and control structures in simplifying rational expressions 2). solve rational equations by multiplying both sides by the LCD to eliminate the denominator. Â Students will be required to work in cooperative learning groups and their solutions which includes a planned pseudo code, the source code and the output will be presented to the class demonstrating that the program meets the goal of the assignment.
Programming Task 2
Students will write and run a program that uses rational equations to solve rate problems and distance estimations of the projectile motion or other motions like Indy car race and fluid flow. Students will be required to work in cooperative learning groups and their solutions which includes an a planned pseudo code, the source code and the output will be presented to the class demonstrating that the program meets the goal of the assignment.
Programming Task 3
Skill Practice sets from State Approved Algebra 2 Textbook
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Is Algebra 2 Really That Hard
After going through this post, it may seem like it will be hard to do well in Algebra 2. However, just through the understanding that it is just an extension of concepts that have already been learned, the hardest part of the class is finding students weaknesses from past math classes and addressing those so that they do not have to struggle with these basics when answering harder questions in Algebra 2. Once these bases are covered, it will be much easier for a student to feel comfortable with the material that is additive from Algebra 1 as well as to be ready to understand the new concepts introduced.
High School Algebra 2 Curriculum
Students typically learn Algebra II in 11th grade. An Algebra II curriculum usually builds on knowledge and skills that are gained in Algebra 1 and reinforced in Geometry, including relationships between quantities through equations and inequalities, graphing of functions, and trigonometry.
In high school, Algebra 2 helps students gain an understanding of statistics and probability, exponents and logarithms, and mathematical modeling. In general, the Algebra 2 course covers components in four critical areas: functions, collecting and analyzing data, periodic phenomena, and polynomials.
Learn how to help your Algebra 2 student find success and why Time4Learnings online algebra 2 course for high school may be the right choice for you:
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Why Do Students Find Algebra 2 So Hard
As previously discussed, Algebra 2 is perceived to be hard because it builds on and combines material from many previous math classes, including Algebra 1. This combination exposes weaknesses in students understanding of concepts from those past math classes that need to be practiced and strengthened before being able to understand concepts and do well in algebra 2. For example, a student will have a hard time grasping solving polynomial functions if they struggle with applying order of operations, using fractions or do not know how to solve linear equations, subjects taught early in math learning and in algebra 1. Math is one of the only subjects where learning is incremental, meaning that the concepts introduced in previous math classes are necessary for success in subsequent classes, which use these fundamentals through different applications, and this idea is truly exposed in Algebra 2.
Algebra Ii Just Doesnt Add Up When You Figure How Little It Means To Most Students
I got an A in algebra II, I think. That was long ago. I do know that I have long since forgotten whatever I learned in that course and have never used it since.
That has become a national problem. Algebra II is required for graduation in 20 states and the District. Yet many experts want to discard it in favor of something more fashionable. These days, they say, students need to understand big data, a course often called statistics.
Algebra II is frequently combined with trigonometry in the third year of high school math. It covers linear equations, functions, exponential and logarithmic expressions, and other things. It became a regular part of American education after high school math was overhauled in the wake of the Soviet Unions launch of Sputnik in 1957.
It no longer gets much respect. The Freakonomics Radio podcast, in a special episode hosted by University of Chicago economist Steve Levitt, surveyed listeners about math subjects they used in their daily lives. Algebra II wasnt part of that poll, but 70 percent of respondents said they never used its close cousin trigonometry.
That includes me. I use math in my work but only the long division I learned in fourth grade. It helps me prepare my annual list of schools with high rates of college-level test preparation. A calculator I got free in the mail from SPCA International does the actual arithmetic for me.
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Learning Objectives For Algebra Ii
At this level, your childs goals and objectives for Algebra 2 may be very specific, targeting discrete concepts or skills. Here are some examples of 11th grade Algebra 2 learning goals and objectives:
- Evaluate sums, differences, products, and quotients of functions.
- Use the discriminant to determine the number and type of roots of a quadratic equation.
- Determine the roots of and factor a polynomial function.
- Graph rational functions that have only vertical or horizontal asymptotes.
- Write one-variable radical inequalities to model problems.
- Solve logarithmic expressions by converting between logarithmic and exponential forms.
- Compare sine, cosine, and tangent values for angles having the same reference angle.
- Analyze a function rule or graph to determine transformations of the parent function.
Should I Skip Algebra 2 And Take Pre
HI. So I am a freshmen and i just took geometry in 9th grade. Honestly it was easy for me. So I was thinking if is should skip Algebra 2 for 10th grade, like should i study the entire course during the summer and when i come back to school study pre-calc. I plan on doing 10th grade: Pre-calc 11th: Ap Calc BC 12th: AP stats? or anything else
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Unit : Systems Of Equations
The assignments for this unit require the student to demonstrate mastery with the Algebra 2 topic of solving systems of equations. Initially, the student will solve systems by the traditional paper and pencil approach. This will lead the student to develop an algorithm that can then be used when writing to program. This meets the UC “c” requirement for the traditional Algebra 2 course along with the CTE goal of programming using 2-D arrays, transitioning from the use of 1-D arrays in the previous unit.
Solve without Technology
Students complete a series of systems of equations problems using algebraic methods and check their solution by graphing the system. The problems will be selected to encourage use of elimination method which is the basis of solving using matrices.
Solve with Technology
Students solve systems of equations with graphing calculator using matrices. Students will learn about the format of augmented matrices in order to solve using row reduced echelon form. Upon completion of these problems, students will be ready to start their program.
Students write a program to solve a system of equations in two variables. The students will use the 2-dimensional array to store the equations and manipulate the data based on the elimination method and resulting in a row reduced echelon form matrix. Once their program is proven effective by checking it against their previously solved problems, they extend it to larger systems.