ALGEBRA. Expressions An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general terms, abstracting from specific instances. Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor. Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure. A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave. Equations and Inequalities An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form. The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system. An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to extran...
ALGEBRA. In this annex the pedagogical contents and challenges of the Algebra Lesson Plan are fully described. This is a basic example of the contents that can be “slotted” into Game Plots, and relies mostly on minigames and quizzes to provide evidence and progress. Basic algebraic skills refresher course
ALGEBRA. For any abelian group M we set Mp := Zp ⊗Z M and write Mp∧ for its pro-p completion ←− lim n M/pnM (so that Mp = Mp∧ if M is finitely generated). For any natural number n we write M [n] for the subgroup of M comprising those elements that have order dividing n. We write Mtor for the union of M [n] over all natural numbers n, set M := M/Mtor and identify M with a sublattice of the vector space Q ⊗Z M in the obvious way. If M is a finitely generated Zp-module we define its ‘Zp-rank’ to be rkZp (M ) := dimQp (Qp ⊗Zp M ) and its ‘p-rank’ to be rkp(M ) := dimFp (M/pM ). We note that rkp(M ) = dimFp (M [p]) +rkZp (M ) and that for any exact sequence of finitely generated Zp-modules M1 −θ→1 M2 −θ→2 M3 one has rkp(Im(θi)) ≤ rkp(M2) ≤ rkp(M1) + rkp(M3) for both i = 1 and i = 2. We often abbreviate the vector space Qp ⊗Zp M to Qp ·M . If Γ is a finite group, then a finitely generated Zp[Γ]-module M is said to be a ‘Zp[Γ]- lattice’ if Mtor vanishes. Each Zp[Γ]-lattice M can therefore be regarded as a submodule of the space Qp ·M .
ALGEBRA. (One of the following) 1. Algebra II in high school with at least a C- grade. 2. MATH 096 Intermediate Algebra MATH 151 Elementary Algebra
ALGEBRA. Knowledge of the central issues of algebra up to and including the equations of the second degree with one unknown variable. Geometry: Knowledge of the central issues of two and three-dimensional geometry. (Xxxxxxx, 1979, p. 19) As a result of the New Training College Act of 1952, teacher training was changed. The school subjects of secondary education were replaced with the teaching methods for the subjects taught in primary schools. For example, mathematics was replaced with teaching methods for arithmetic (Xxx Xxxxxx, 1964). However, because the teacher educators remained the same, little changed in practice. The teaching methods for arithmetic were frequently augmented with tough calculations for the student teachers, supplemented with some hints for working in the classroom. Most of these hints were of a general educational nature, they referred to for example teaching with visual aids, and elements of educational psychology such as different levels of thinking (Xxxxxxx & Xxxx, 1999). Although theory and practice were closer in terms of curriculum development, subjects remained isolated and methods were far from the real practice. Xxxxxxx (1979) gives an example of a mathematics method book for teacher education the title of which, ‘Theory and Practice,’ raises high expectations. In the explanation by the author X.X. Xxxxxx (1963), it turns out that his concept of ‘theory’ encompassed the art of arithmetic and that ‘practice’ implied skills of arithmetic. This belief is characteristic for the lack of vision in teacher education during the 1950s and 1960s. The reorganization of 1968, when teacher-training colleges were to be named by law ‘Pedagogical Academies,’ did not in general lead to important changes. Because of the idea that the methods of ‘all subjects of the primary school’ should be taught, the matter of domain-specific instructional theory barely existed. As a result, teacher-training curricula remained fragmentary, with consistency and commonality towards goals lacking. In fact, ‘theory’ for the student teachers comprised mainly general educational theory. In 1984, the teacher training colleges were reorganized to a four year course, but it was only in the early 1990s that any significant changes occurred. Research by among others Xxxxxx (1987) and Corporaal (1988) indicates that the desired consistency between theory and practice was still absent. While there was a shift towards practice, the need to do so did not necessarily arise from mo...
ALGEBRA. For a unital ring A, we write A× for the multiplicative group of invertible elements of A and ζ(A) for the centre of A. By an A-module, we shall always mean a left A-module. For the inclusion of rings A ⊂ A′, and an A-module M , we write MA' := A′ ⊗A M . For each pair of A-modules M and M′ we write IsA(M, M′) for the set of A-module isomorphisms from M to M′ and AutA(M ) for the group of A-module automorphisms of M . Fix a finite group Γ, and a Dedekind domain R of characteristic zero, with field of fractions F , and E is an extension field of F . We let OE denote the ring of integers of a field E. For any R[Γ]-module M , we write ME := E⊗RM , and Mv := Rv ⊗RM , with Rv the completion of R at v, where v is a (non-zero) prime ideal of R. Let E/F be a finite Galois extension of fields. We let Gal(E/F ) denote the Galois group of E/F , and we write the action of Gal(E/F ) on E by x '→ g(x) for x ∈ E and g ∈ Gal(E/F ). We fix a separable closure Fc of F , and write ΩF for the absolute Galois group Gal(Fc/F ) of
