Rewriting Formulas and Equations

5 Key excerpts on "Rewriting Formulas and Equations"

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  eBook - ePub

  Maths from Scratch for Biologists

  • Alan J. Cann(Author)
  • 2013(Publication Date)
  • Wiley

    (Publisher)

  2

  Manipulating Numbers

  Algebra (from the Arabic, al-jabr, ‘the reduction’) – a form of maths where symbols are used to represent numbers

  LEARNING OBJECTIVES:

  On completing this chapter, you should be able to:

  • understand the basic rules of algebra;
  • perform simple algebraic manipulations;
  • identify and manipulate fractions.

  Arithmetic is concerned with the effect of operations (e.g. addition, multiplication, etc.) on specified numbers. In algebra, operations are applied to variables rather than specific numbers. Why? Here is a classic example:

  John is 10 years old. His father is 35 years old. After how many years will the father be twice as old as the son?

  You could try to find the answer by experimenting with different numbers, but this is laborious. The better way is to treat this an algebra problem and write the problem as an equation which we can then solve.

  Let the father be twice as old as the son in x years time. The son will then be (10 + x ) years old and the father will be (35 + x ) year old:

  Therefore,

  Simplify this by subtracting x from each side to keep the equation balanced:

  Simplify by subtracting 20 from each side to keep the equation balanced:

  2.1. Manipulating numbers

  To manipulate numbers, you need to know the rules. In mathematics, this is known as the ‘Order of Operations’ – an internationally agreed set of arbitrary rules which allows mathematicians the world over to arrive at the same answers to problems:

  In algebra, there are two sorts of statement which you need to be able to recognize:

  1. A mathematical expression is a string of symbols which describes (‘expresses’) a (potential) calculation using operators (symbols indicating an operation to be performed, e.g. plus, minus, divide, etc.) and operands (symbols which the operators act on), e.g.

  Expressions do not contain an equal sign, but can often be simplified, that is converted to a simpler form containing fewer terms. 2. A mathematical equation contains an equal sign. The terms (groups of numbers or symbols) on both sides of the equal sign are equivalent, e.g.

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  eBook - ePub

  Routledge Revivals: Speaking Mathematically (1987)

  Communication in Mathematics Clasrooms

  • David Pimm(Author)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  The meta-language of arithmetic is algebra. Many, but not all, of the transformations – the laws of arithmetic – are either taught explicitly to the pupils in the meta-language, or in the form of arithmetic specializations from which the pupils are expected to extrapolate the general form. There remains the ambiguity of whether these are the laws of arithmetic or the laws of algebra.
• Transformations are often reversible, but are frequently applied only in a left-to-right manner and mutually-inverse pairs are perceived and described differently. For instance, if the notion of phrase is extended to include letters, thereby switching from the context of arithmetic to that of algebra itself, then:
  Expanding brackets:

  (x + a )(x + b ) →

  x2

  + ax + xb + ab (This is now the surface form in algebra.)

  and Factoring:

  x 2 + (a + b)x + ab → (x + a)(x + b )

  are seen as two separate actions, and hence, transformations.
• The prevalent language of the teacher in describing such transformations is often purely in terms of the surface structure, thereby focusing attention on the form rather than the meaning which gives rise to the transformation.
  • Take it over to the other side and change the sign.
  • Cross multiply.
  • Move the decimal point over.
  • Turn it upside down and multiply.
  • Collect all the xs on one side of the equation.
  • Always do to the top what you do to the bottom.
  • To multiply by ten add a nought.
  All of these descriptions are to do with actions on the symbols, on how to achieve grammatical utterances from other ones. They tell what should be done, with the result that there is little impetus to examine them to see why they might be helpful transformations to carry out. Mathematically interesting things can occur if they are not blindly followed on every occasion. See Brown and Walter (1983) for an exploration of one outcome of ignoring the injunction to ‘Collect all the xs on one side of the equation.’
• Mathematical syntactic rules are prescriptive and are consciously and deliberately taught, learned and applied at the surface level. When confronted with a student who had written

  ∫edx

  , I heard myself say ‘You never have a dx