Fractions:
key ideas and important concept knowledge
Key ideas
The overarching key ideas of estimation, benchmarks, visualisation, equality and equivalence, language and strategies need to be considered when developing units of work in fractions. The specific key ideas in fractions are: quantity, number triad, partitioning, equivalence and benchmarks.
Quantity
A quantity is an amount of something which is determined using a number and a unit.
Number triad
A number triad represents the relationship between words, symbols and materials or diagrams of a quantity.
Partitioning
A quantity can be separated into parts while maintaining a sense of the whole.
Equivalence
Benchmarks
Benchmarks are trusted quantities or numbers used as reference points to estimate, calculate or compare.
Important concept knowledge
Meaning of the symbols
In any fraction, the top number is called the numerator and the bottom number is called the denominator. The horizontal line that separates the numerator from the denominator is called the vinculum. The denominator indicates the size of the parts. The numerator indicates the number of parts of that size.
Fraction types
Proper fraction
Unit fraction
Improper fraction
Mixed number
Iteration
Iteration is a repeated copying of a unit with no gaps or overlaps to form a quantity.
Fraction sub-constructs
Fraction sub-constructs are the work of Tom Kieren (Kieren 1980, 1988, 1993) and are detailed in Table 3.
Table 3: Fraction sub-constructs
Strategies for ordering fractions by size
Residual thinking
Residual thinking is using the ‘left over’ amount when two or more fractions are compared to one, or another benchmark like one-half.
Converting to equivalent fractions with the same denominator
Converting to equivalent fractions with the same denominator refers to renaming both fractions so the denominators are the same.
Representations
Physical and diagrammatic representations can be discrete or continuous: discrete representations involve collections of objects; and continuous representations can be partitioned anywhere to create fractions, and include lengths, area, volumes or capacities and mass.
Misconceptions
Most incorrect ideas students possess are the result of overgeneralising the properties of whole numbers and transferring those properties to rational numbers. Below are some examples of common misconceptions.
Ordering by numerators
Given that the numerator tells how many parts, students may incorrectly order fractions by the numerators.
Ordering by denominators
Ordering by denominators (or reciprocal thinking) refers to students incorrectly believing that fractions can be ordered by finding the smaller denominator. This
misconception arises since the more equal parts a whole is cut into, the smaller the parts become.
Gap thinking
Gap thinking refers to comparing non-unit fractions (i.e. a fraction with a numerator greater than 1) by considering the number of parts rather than the size of the part.
Adding numerators and denominators
Given that fractions have whole numbers as numerators and denominators, some students think that addition of fractions works like whole numbers, incorrectly adding numerators together and denominators together.
Percentages
A percentage is a fraction with a denominator of 100. The literal meaning of the % sign is 'per hundred' which comes from the vinculum (line) of a fraction combined with the two zeros from 100.
Applications of percentages
Often percentages refer to a part–whole ratio.
Percentages can be greater than 100, where they represent a comparison of two quantities.
Percentages also act as operators.
Strategies
Many strategies are useful for calculating with percentages. Some examples include:
• represent percentages as equivalent part–whole ratios using a dual number line
• use 10% as a benchmark
• convert percentages to simple fractions
• find the unit rate (1%) and multiply (unitary method)
• use common factors to simplify the ratio.
