8 Algebraicexpressions Equations Lessonplan Vinay

Instructional Objectives

(This plan is till mid august)

  • An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.
  • The equations we study in the previous classes are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.
  • A linear equation may have for its solution any rational number.
  • An equation may have linear expressions on both sides.
  • Just as numbers, variables can, also, be transposed from one side of the equation to the other.
  • Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.
  • The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved using linear equations.
  • Expressions are formed from variables and constants.
  • Terms are added to form expressions. Terms themselves are formed as product of factors.
  • Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively. In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.
  • Like terms are formed from the same variables and the powers of these variables are the same, too. Coefficients of like terms need not be the same.
  • While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.
  • There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.
  • A monomial multiplied by a monomial always gives a monomial.
  • While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the monomial.
  • In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial). Note that in such multiplication, we may get terms in the product which are like and have to be combined.
  • An identity is an equality, which is true for all values of the variables in the equality. On the other hand, an equation is true only for certain values of its variables. An equation is not an identity.
  • The following are the standard identities:
      • (a + b)2 = a2 + 2ab + b2 (I)
      • (a – b)2 = a2 – 2ab + b2 (II)
      • (a + b) (a – b) = a2 – b2 (III)
      • (x + a) (x + b) = x2 + (a + b) x + ab (IV)
  • The above four identities are useful in carrying out squares and products of algebraic expressions. They also allow easy alternative methods to calculate products of numbers and so on.
  • When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.
  • An irreducible factor is a factor which cannot be expressed further as a product of factors.
  • A systematic way of factorising an expression is the common factor method. It consists of three
    • Write each term of the expression as a product of irreducible factors
    • Look for and separate the common factors
    • Combine the remaining factors in each term in accordance with the distributive law.
  • Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.
  • In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.
  • A number of expressions to be factorised are of the form or can be put into the form : a2 + 2 ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b) + ab. These expressions can be easily factorised using Identities
    • a2 + 2 ab + b2 = (a + b)2
    • a2 – 2ab + b2 = (a – b)2
    • a2 – b2 = (a + b) (a – b)
    • x2 + (a + b) x + ab = (x + a) (x + b)
  • In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab. Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.
  • We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.
  • In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.
  • In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.
  • In the case of divisions of algebraic expressions that we studied in this chapter, we have
    • Dividend = Divisor × Quotient.
    • In general, however, the relation is Dividend = Divisor × Quotient + Remainder
    • Thus, we have considered in the present chapter only those divisions in which the remainder is zero.

Teaching Process

  • Multiplication and division of algebraic exp.(Coefficient should be integers)
  • Some common errors (e.g. 2 + x ≠2x, 7x+ y ≠7xy)
  • Basic operations on algebraic expressions
  • Identities (a± b)2 = a2 ± 2ab+ b2, a2 – b2 = (a– b) (a + b)
  • Factorisation (simple cases only) as examples the following types a(x+ y), (x± y)2 , a2 – b2 , (x+ a).(x+ b).
  • Solving linear equations in one variable in contextual problems involving multiplication and division (word problems) (avoid complex coefficient in the equations)
  • Reducing equations to simpler forms and linear forms

Special needs

  • Catch up worksheets and IEP

Evaluation tools

  • Constant evaluation through interactions in classroom.
  • Seminar by students on a self chosen topic
  • Exercise questions and worksheets on respective sub topic covered.
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