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Algebra I
CLASS OFFERINGS
LESSON 1
Briefly stated, algebra is the study of how numbers behave, and it is used to solve real-life problems.
However, if you don't know the different types of numbers that exist, learning which numbers behave in exactly what kinds of ways is likely to be more difficult.
Accordingly, our first presentation will begin familiarizing you with the five main categories of numbers with which you will be working.
1. Imaginary
2. Real
3. Rational
4. Irrational
5. Integers
Generally speaking, numbers are divided into two major groups: real numbers and imaginary numbers. We are supposed to tell you that real numbers are numbers found on the number line, but we do not think this very helpful.
What is helpful however is knowing how to tell the two types of numbers apart. You see, they are easily distinguished from one another because imaginary numbers are always accompanied by an italicized lowercase letter “i.” It is therefore safe to assume any number which is not accompanied by an italicized lowercase letter “i” is a real number.
The real numbers are further divided into two subsets: rational numbers and irrational numbers.
A rational number is basically any fraction consisting of integers (numbers that represent complete units, along with their corresponding negatives, and zero). Note that all rational numbers have either terminating or repeating decimals.
Also, keep in mind that, since any integer can be turned into a fraction simply by placing it over a fraction bar and the numeral one, all integers are rational numbers. The integers include natural numbers and whole numbers as described by Allison Moffet.
Unlike rational numbers, irrational numbers have never-ending non-repeating decimals and cannot be expressed as fractions. Typical examples are pi, the square root of two, and e.
Here is a quick summary reviewing most of the above information.
And this lesson from the "Video Math Tutor" covers all of the above material in far greater detail.
LESSON 2
We have included this video clip to ensure you are provided with at least one formal lesson on set builder notation.
These lessons go into set theory in more detail:
We have not yet reviewed this entire lesson in that it is almost an hour long. However, if you would like to watch a presentation in which the information is introduced or taught more slowly, this may be more to your liking.
LESSON 3
Now that you know about the various number sets (or number systems) it's time for you to start becoming familiar with their corresponding properties such as closure, indentity, inverse, etc.
This presentation also describes and explains properties and laws of numbers to help you deepen your understanding of the real number system and equip you with some of the basic knowledge required to grasp important algebraic concepts.
LESSON 4
Now that you are familiar with the types of numbers and their various laws and propreties, let's have you begin practicing how to identify and use them where applicable.
We'll start by havin you use properties of numbers to demonstrate whether assertions are true or false.
LESSON 5
Exponents are usually taught sporadically with instruction delivered helterskelter throughout the duration of the school year.
However, given that we expect our students to be able to list and describe all ten rules in an organize fashion and thus, that is the way we teach them.
These additonal lessons are offered to further help you understand and use such operations as taking the opposite, finding the reciprocal, taking a root and raising to a fractional power.
LESSON 6
3. Students solve equations and inequalities involving absolute values.
LESSON 7
4. Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
LESSON 8
5. Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
LESSON 9
6. Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
LESSON 10
7. Students verify that a point lies on a line, given an equation on the line. Students are able to derive linear equations by using the point-slope formula.
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8. Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
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9. Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
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10. Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these techniques.
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11. Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a
polynomial, recognizing the difference or two squares, and recognizing perfect squares of binomials.
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12. Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
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13. Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
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14. Students solve a quadratic equation by factoring or completing the square.
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15. Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
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16. Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
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17. Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
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18. Students determine whether a relation defined by a graph, a set of ordered pairs, or symbolic expression is a function and justify the conclusion.
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19. Students know the quadratic formula and are familiar with its proof by completing the square.
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20. Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
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21. Students graph quadratic functions and know that their roots are the x-intercepts.
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22. Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
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23. Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
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24. Students use and know simple aspects of a logical argument.
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24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
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24.2 Students identify the hypothesis and conclusion in logical deduction.
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24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
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25. Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements.
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25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
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25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.
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25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expression or equations or inequalities, students determine whether the statement is true sometimes, always, or never.
In addition, teachers will incorporate the standards from Communication in Mathematics and Logical/Mathematical Reasoning Across the Curriculum in every mathematics course.
