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- 1. 5.statement that two algebraic expressions are equal inequality equation variable polynomial 6.equation whose terms contain squares of the variable and no higher powers quartic quadratic cubic linear
- 2. MAERK CHECK (/) FOR THE CORRECT ANSWER1.number that multiplies a variable or variables monomial - binomial - coefficient Why is Algebra so Important? - trinomial It was not very long ago when good2. computation skill in arithmetic was the keya number, a variable or product of numbers and to getting a job, a diploma, or managing a variables monomial business. Even as late as the 1970s we expected that most students who finished - binomial high school would be able to calculate - coefficient quickly and accurately. We taught algebra - trinomial and geometry to our college bound, and3. just a very few of the math geeks tookletter that stands for an unknown number constant calculus. - coefficient Today the world has radically changed. - binomial Algebra has moved down to the 7th and - variable 8th grades for many of our students and it 4. is a requirement for high schoola monomial or a sum of two or more graduation, for college graduation, and for monomials unlike terms most any job in America today. But is this - like terms right? Should we be demanding algebra as a rite of passage? - polynomial - equation The answer is a resounding yes!
- 3. Algebra is not only the essential language of mathematics it is about two ideas that are critical to 21st century jobs and citizenship. The first idea is variable. Variables are symbols that can represent not only a number but a quantity that is changing. A variable can represent a ball being thrown or an automobile being driven or the Dow Jones Average. Algebra lets us manipulate variables like arithmetic lets us manipulate numbers. The second fundamental idea, one that is often called the most important idea in mathematics, is the concept of function. A function is a well defined relationship between two variables so that as the value of one changes so does the value of the other. With variables and functions we can command spreadsheets, we can define the physical laws that govern our universe, and we can build patterns that enable us to understand how our world works.
- 4. This is the solution to algebra number problem 13 as asked by an anonymous user: "A maths test So algebra is more than a set of rules and contains 10 questions. Ten points are given for procedures for solving canned problems. It each correct answer and three points deducted for is a way of thinking. We built Enablemath an incorrect answer. If Ralph scored 61, how many to be did he get correct?" algebraic. Even in the simplest assignments we have students think Ok on this problem you can come up with a algebraically. Students can not only step solution faster by just quick trial an error. You through an example, they can change the know that if it only counts 3 points off for each values, treating numbers as variables. wrong answer and they got a 61 then they had to They can see how these values are related have at least a 70 before the points were taken to each other in the dynamic visualizations off. If they got a 70 that means they missed 3 that make our presentation of concepts problems (3 * 3 = 9) 70 - 9 = 61 . So the answer understandable. The What if Wheel gives is they got 7 problems right and 3 wrong. every student the power over Now to set this up using algebra... variables, and the screens are populated 10x - 3(10-x) = 61 You are subtracting the wrong with objects that have functions tied to answers worth 3 points a piece from the right these variables. ones worth 10 a piece. You dont know how many Our students not only learn the algebra of each so you are saying there are x amount of that they need for school and for passing 10 valued answers and (10 - x) number of 3 exams, they learn and intuitively valued wrong answers. understand algebra and use the 10x - (30 - 3x) = 61 understanding to apply it to any problem 10x -30 + 3x = 61 that they may find. We believe that no one 13x = 91 --> x = 91/13 = 7 --> so there are 7 else has done this in as rich and right 10 valued problems and (10-7) or 3 wrong 3 comprehensive a fashion. Yes, algebra is point negative value ones..... fundamental, but it also has to be learned in a new way. We have created You could have also set it up like this 10(10 - x) - Enablemath with that in mind. We believe
- 5. Elementary algebra Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of mathematicsbeyond the basic principles of arithmetic. In arithmetic, only numbers and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often denoted by symbols (such as a, x, or y). This is useful because: It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system. It allows the reference to "unknown" Across Down numbers, the formulation of equations and 1. 10 + 5 2. 79 -28 the study of how to solve these. (For 3. 52 + 32 4. 30 + 15 instance, "Find a number x such that 3x + 1 6. 18 - 7 5. 54 - 10 = 10" or going a bit further "Find a 8. 97 - 40 7. 21 - 9 number x such that ax+b=c". This step leads to the conclusion that it is not the 10. 51 - 9 9. 98 - 28 nature of the specific numbers that allows 12. 47 - 26 11. 16 + 8 us to solve it, but that of the operations 14. 32 + 14 13. 23 - 10 involved.) 16. 78 - 47 15. 18 +43 19. 9 + 7 17. 26 -12 It allows the formulation 21. 50 - 9 18. 48 + 4 of functional relationships. (For instance, "If 23. 70 - 42 20. 78 - 15 you sell x tickets, then your profit will be 3x − 10 dollars, or f(x) = 3x − 10, where f is 25. 79 - 47 22. 13 - 3 the function, and x is the number to which 27. 4 + 21 24. 78 + 4 the function is applied.") 28. 52 - 29 26. 50 - 28
- 6. Example #1 Example Polynomials + #2 - A polynomial is an expression that is =? constructed from one or more variables and constants, using only the operations of =? addition, subtraction, and multiplication (where repeated multiplication of the same variable is standardly denoted as exponentiation with a constant nonnegative integer exponent). For example, x2 + 2x − 3 is a polynomial in the single variable x. An important class of problems in algebra is factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials. The example polynomial above can be factored as (x − 1)(x + 3). A related class of problems is finding algebraic expressions for the roots of a polynomial in a single variable.
- 7. Abstract algebra Abstract algebra extends the familiar concepts found in elementary algebra and arithmetic of numbers to more general concepts. Adding or Subtracting Sets: Rather than just considering the different types of numbers, abstract algebra deals with the Rational Expressions more general concept of sets: a collection of all with Like Denominators objects (called elements) selected by property, specific for the set. All collections of the To add or subtract rational familiar types of numbers are sets. Other expressions with like examples of sets include the set of all two-by- denominators, add or two matrices, the set of all second- degree polynomials (ax2 + bx + c), the set of all subtract their numerators two dimensional vectors in the plane, and the and write the result over various finite groups such as the cyclic the denominator. groups which are the group of integers modulo n. Set theoryis a branch Then, simplify and factor of logic and not technically a branch of algebra. the numerator, and write Binary operations: The notion of addition (+) is the expression in lowest abstracted to give a binary operation, ∗ say. The terms. This is similar to notion of binary operation is meaningless without the set on which the operation is defined. For two adding two fractions with elements a and b in a set S, a ∗ b is another like denominators, as in. element in the set; this condition is called closure. Addition (+), subtraction (- ), multiplication (×), and division (÷) can be binary operations when defined on different sets, as is addition and multiplication of
- 8. Rational Expressions Identity elements: The numbers zero and When we discuss a rational expression in this one are abstracted to give the notion of chapter, we are referring to an expression whose an identity element for an operation. Zero is numerator and denominator are (or can be written the identity element for addition and one is as) polynomials. For example, and are rational the identity element for multiplication. For a expressions. general binary operator ∗ the identity element e must satisfy a ∗e = a and e ∗ a = a. To write a rational expression in lowest This holds for addition as a + 0 = a and 0 + a = a and multiplication a × 1 = a and 1 terms, we must first find all common factors × a = a. Not all set and operator combinations (constants, variables, or polynomials) or have an identity element; for example, the the numerator and the denominator. Thus, positive natural numbers (1, 2, 3, ...) have no we must factor the numerator and the identity element for addition. denominator. Once the numerator and the Inverse elements: The negative numbers denominator have been factored, cross out give rise to the concept of inverse elements. any common factors. For addition, the inverse of a is −a, and for Example 1: Write n lowest terms. multiplication the inverse is 1/a. A general inverse element a−1 must satisfy the property Factor the numerator: 6x 2 -21x - 12 = 3(2x 2 - 7x - 4) that a ∗ a−1 = e and a−1 ∗ a = e. = 3(x - 4)(2x + 1) . Associativity: Addition of integers has a property called associativity. That is, the Factor the denominator: 54x 2 +45x + 9 = 9(6x 2 + grouping of the numbers to be added does 5x + 1) = 9(3x + 1)(2x + 1) . not affect the sum. For example: (2 + 3) + 4 = Cancel out common factors: = 2 + (3 + 4). In general, this becomes (a ∗ b) ∗ c = a ∗ (b ∗ c). This property is shared by most binary operations, but not subtraction or Example 2: Write in lowest terms. division or octonion multiplication. Factor the numerator: x 3 - x = x(x 2 - 1) = x(x + 1)(x - 1) . Commutativity: Addition and multiplication of 4 3 Factor the denominator: 6x +2x -8x = 2x 2 2(3x 2 + x - 4) = real numbers are both commutative. That 2x 2(x - 1)(3x + 4) . is, the order of the numbers does not affect Cancel out common factors: = . the result. For example: 2+3=3+2. In general, this becomes a ∗ b = b ∗ a. This property does not hold for all binary operations. For example,matrix multiplication and quaternion multiplication are both non-commutative.
- 9. Groups Combining the above concepts gives one of the most important structures in mathematics: a group. A group is a combination of a set S and a single binary operation ∗, defined in any way you choose, but with the following properties: An identity element e exists, such that for every member a of S, e ∗ a and a ∗ e are both identical to a. Every element has an inverse: for every member a of S, there exists a member a−1 such that a ∗ a−1 and a−1 ∗ a are both identical to the identity element. The operation is associative: if a, b and c are members of S, then (a ∗ b) ∗ c is identical to a ∗ (b ∗ c). If a group is also commutative—that is, for any two members a and b of S, a ∗ b is identical to b ∗ a—then the group is said to be abelian. For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element a is its negation, −a. The associativity requirement is met, because for any integers a, b and c, (a + b) + c = a + (b + c) The nonzero rational numbers form a group under multiplication. Here, the identity element is 1, since 1 × a = a × 1 = a for any rational number a. The inverse of a is 1/a, since a × 1/a = 1. The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is ¼, which is not an integer. The theory of groups is studied in group theory. A major result in this theory is the classification of finite simple groups, mostly

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