Fractions for Algebra and Arithmetic: 4.5 - Part 1

Supporting Materials

• Related progression points
• Developmental Overview of Methods of Calculation (PDF - 38Kb)
• Developmental Overview of Structure (PDF - 94Kb)

 

Indicator of Progress

Success with using fractions and fraction-like expressions in algebra depends on four key understandings:

• Students recognise that in algebra, fraction notation represents the operation of division
• They can deal with factorised expressions and know what to do with the common factors)
• They understand that the fraction vinculum (the line between the numerator and denominator in a fraction) acts as a set of brackets for both the numerator and denominator and can act on the implications for calculation
• They use their knowledge of fraction multiplication and division to deal with fraction-like expressions

 

Fraction notation represents division (provided y is not 0)

 

Dealing with factorised 'unclosed' expressions and common factors

 

Fraction vinculum acts as a set of brackets

 

Fraction-like expressions need fraction multiplication and division

 

 

The focus here is on good facility with complex fractions and fraction-like expressions. The ability to multiply and divide fractions and to factorise whole numbers are required number skills at this level.

Fractions are important as rational numbers, but they are also important because fraction notation is the way in which division is written in algebra, where the division sign ÷ is not used. Facility with fractions is needed for many aspects of algebra including algebraic fractions (i.e. any division), rationalising surds and solving equations.

Illustration 1: Errors in cancelling

• Students may not understand the structure of a fractional expression. This can result in errors, such as Example 1a below, if they don’t appreciate that the fraction bar (vinculum) automatically brackets the numerator and denominator.
• Students may incorrectly cancel when addition or substraction is involved, e.g. students may cancel the 2 as in Example 1b.
• Lack of understanding of structure of expressions may lead to students cancelling a power or base. For example, students may ‘cancel the 2’ as in Example 1c, or may ‘cancel the 3’ as in Example 1d. Alternatively students may incorrectly cancel a base as in Example 1e. These errors are readily eliminated if students write out the factors in full (e.g. 5² = 5 × 5)

Illustration 2: Quirky quotients

In algebra, fraction notation is used for division. Consequently, ‘fractions’ arise that appear unusual and perturbing to students.  Can students see the main structure (and hence which number is divided by which) in unusual fraction-like expressions like these?

Illustration 3: What is an equivalent fraction anyway?

Tradition has it that we always talk about ‘equivalent fractions’. Some students may not know that ‘equivalent fractions’ are simply ‘equal fractions’. Fraction algorithms only make sense with this understanding.

Teaching Strategies and Further Resources

For Fractions for Algebra and Arithmetic teaching strategies and further resources, see: Part 2