Decimals:
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 decimals. The specific key ideas in decimals are: quantity, number triad, equivalence, partitioning, base-10 system, digit position 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.
Equivalence
A decimal can be equivalent to a fraction or a percentage; for example, a decimal can be expressed as:
• a fraction – 0.5 is the same as one-half
• a percentage – 0.5 is the same as 50%.
Partitioning
A quantity can be separated into parts while maintaining a sense of the whole.
• renamed decimal fractions
• renaming and regrouping
• compact and expanded decimal forms.
Base-10 system
The base-10 system is a number system that is based on grouping quantities in tens or partitioning equally into 10 equal parts.
Each place has a value that is 10 times greater than the place to its right, and one-tenth of the value of the place to its left.
Digit position
The place of a digit in a number determines its value.
There are three ways to interpret a single digit within a decimal fraction. They are face value, place value and total value.
Zero can be a placeholder and a digit representing a quantity.
Benchmarks
Benchmarks are trusted quantities or numbers used as reference points to estimate, calculate or compare.
Important concept knowledge
Ragged decimals
According to Roche (2005), ragged decimal fractions have a varying number of digits to the right of the decimal point.
Misconceptions
As occurs with fractions, most misconceptions arise when students overgeneralise the properties of whole numbers and transfer those properties to decimals. Three possible misconceptions that students can have about decimals include: longer is larger, shorter is larger and those who think in terms of money.
Longer is larger
Longer is larger occurs in ‘whole-number thinking’, such as 4.63 is larger than 4.8 as 63 > 8, and ‘column overflow thinking’, such as 4.63 is greater than 4.8 as 63 tenths is greater than 8 tenths.
Shorter is larger
Shorter is larger occurs in ‘denominator-focused thinking’. A student might incorrectly generalise that one-tenth is bigger than one-hundredth, meaning that any number of tenths is bigger than any number of hundredths. For example, 0.4 is bigger than 0.83.
The shorter-is-larger misconception also occurs in ‘reciprocal thinking’. In this case, a student sees the decimal fraction part as the denominator of a fraction, with larger denominators creating smaller fractions. This misconception is revealed when 0.3 is chosen as the larger of 0.3 and 0.4.
In ‘negative thinking’, a student believes 0.3 is larger than 0.4 as -3 is larger than -4.
Money thinkers
Students who are money thinkers have an understanding of the first two decimal places because amounts of money only exist to hundredths of a dollar (cents). They may view decimals as two whole numbers separated by a dot, the first possibly representing dollars and the second cents. It is important to recognise the limitations of teaching decimals through money.