Decimals
Number
Definition
A decimal is a number that has a whole number part and fraction part. The prefix deci comes from the Latin word decimus which means a tenth part.
Students need to become increasingly aware of the use of place value in operating with decimals. Decimals are better suited than fractions to estimating magnitude due to the use of the base-ten system (It is the way we assign place value to numerals. It is sometimes called the decimal system because a digit’s value in a number is determined by where it lies in relation to the decimal point.)
Students can develop misconceptions with decimals stemming from how the base-ten system is used with whole numbers. For instance, students may think that whole numbers with more digits are always larger and hence interpret that 0.75 is larger than 0.8. Students need to understand that larger denominators in fractions result in longer decimals and hence lead to smaller magnitudes. A number with many digits after the decimal point may well be smaller than a number with less decimal digits. So 0.420005 is smaller than 0.5. (National Numeracy Learning Progression)
This unit of work does not include converting decimals into fractions or percentages, and vice versa. Refer to separate units of work in this resource for equivalence within Decimals, Fractions and Percentages.
Professional learning
Professional learning is available to assist teachers to support students in their learning about decimals across the curriculum:
Teaching and learning activities
The resources below provide targeted teaching strategies to support student improvement in this skill.
Each downloadable lesson activity includes:
learning intentions
a list of required resources
a step-by-step lesson sequence
printable classroom materials.
Select the download all icon to download all available activities or select each activity separately.
PLAN2 Areas of focus
An Areas of focus template has been created in PLAN2 to support targeted teaching of Text structure in your learning area.
Search for the DoE template titled ‘DoE HSCMinStd Writing: Text structure’ in the Areas of focus template library tab within the Plan menu, and customise it for your students’ needs.
For more information about using PLAN2 Areas of focus templates with this resource, visit the Using this resource with PLAN2 page.
Relevance to the numeracy test marking
According to the ACSF, the feedback for a Level 3 performance in the HSC minimum standard online numeracy test for operating with decimals states:
Individuals performing at this level typically select appropriate strategies from a variety of everyday mathematical processes in familiar and some less familiar contexts. They interpret and comprehend mathematical information in written material, diagrams, charts and tables as well. They use large whole numbers in words and figures, and understand and convert routine decimals.
Connections with ACSF Level 3 descriptors
The relevant Level 3 ACSF descriptors for numeracy are shown here to demonstrate how operating with decimals are assessed in the HSC minimum standard online test. The performance features identified show what a student is able to do in order to achieve at this level and are provided to support teachers to understand what is required to achieve a Level 3 in this skill.
Numeracy Indicator 3.09: Selects and interprets mathematical information that may be partly embedded in a range of familiar, and some less familiar, tasks and texts.
Focus area: Explicitness and complexity of mathematical information
Level 3 performance features:
calculates with decimals, including division by decimal values.
Numeracy Indicator 3.10: Selects from and uses a variety of developing mathematical and problem solving strategies in a range of familiar and some less familiar contexts.
Focus area: Mathematical knowledge skills: number and algebra
Level 3 performance features:
calculates with decimals, including division by decimal values.
Connections with Numeracy Learning Progression
The progressions describe a typical developmental sequence of literacy and numeracy learning. The numeracy progression sub-elements, levels and indicators relevant to operating with decimals are provided here to assist teachers to identify students’ capabilities and needs to support targeted teaching.
Element: Number Sense and Algebra
Sub-element: Number and place value
NPV6 — Place value
represents and names tenths as one out of 10 equal parts of a whole (e.g. uses a bar model to represent the whole and its parts; uses a straw that has been cut into ten equal pieces to demonstrate that one piece is one-tenth of a whole straw and two pieces are two-tenths of the whole straw)
represents and names one-tenth as its decimal equivalent 0.1, zero point one
extends the place value system to tenths
NPV7 — Numeral recognition and identification
identifies, reads and writes decimals to one and two decimal places
NPV7 — Place value
explains that the place value names for decimal numbers relate to the ones place value
explains and demonstrates that the place value system extends beyond tenths to hundredths, thousandths …
models, represents, compares and orders decimals up to 2 decimal places (e.g. constructs a number line to include decimal values between 0 and 1, when asked ‘which is larger 0.19 or 0.2?’ responds ‘0.2’)
rounds decimals to the nearest whole number in order to estimate answers (e.g. estimates the length of material needed by rounding up the measurement to the nearest whole number)
NPV8 — Numeral recognition and identification
identifies, reads and writes decimal numbers applying knowledge of the place value periods of tenths, hundredths and thousandths and beyond
NPV8 — Place value
compares the size of decimals including whole numbers and decimals expressed to different number of places (e.g. selects 0.35 as the largest from the set 0.2, 0.125, 0.35; explains that 2 is larger than 1.845)
describes the multiplicative relationship between the adjacent positions in place value for decimals (e.g. understands that 0.2 is 10 times as large as 0.02 and that 100 times 0.005 is 0.5)
compares and orders decimals greater than 1 including those expressed to an unequal number of places (e.g. compares the heights of students in the class that are expressed in metres such as 1.50 m is shorter than 1.52 m; correctly orders 1.4, 1.375 and 2 from largest to smallest)
rounds decimals to one and two decimal places for a purpose
NPV9 — Place value
understands that multiplying and dividing numerals by 10, 100, 1000 changes the positional value of the numeral (e.g. explains that 100 times 0.125 is 12.5 because each digit value in 0.125 is multiplied by 100, so 100 x 0.1 is 10, 100 x 0.02 is 2 and 100 x 0.005 is 0.5)
rounds decimals to a specified number of decimal places for a purpose (e.g. the mean distance thrown in a school javelin competition was rounded to two decimal places; if the percentage profit was calculated as 12.467921% the student rounds the calculation to 12.5%)
Element: Number Sense and Algebra
Sub-element: Additive strategies (AdS)
AdS9 — Flexible strategies with fractions and decimals
uses knowledge of place value and how to partition numbers in different ways to make the calculation easier to add and subtract decimals with up to three decimal places
represents a wide range of familiar real-world additive situations involving decimals and common fractions as standard number sentence, explaining their reasoning
Element: Number Sense and Algebra
Sub-element: Multiplicative strategies (MuS)
MuS9 — Flexible strategies for multiplication and division of rational numbers
describes the effect of multiplication by a decimal or fraction less than one (e.g. when multiplying whole numbers by a fraction or decimal less than 1 such as 15 × ½ = 7.5)
connects and converts decimals to fractions to assist in mental computation involving multiplication or division (e.g. to calculate 16 × 0.25, recognises 0.25 as a quarter, and determines a quarter of 16 or determines 0.5 ÷ 0.25, by reading this as one half, how many quarters and giving the answer as 2)
MuS10 — Flexible strategies for working multiplicatively
uses knowledge of place value and multiplicative partitioning to multiply and divide decimals efficiently (e.g. 0.461 × 200 = 0.461 × 100 × 2 = 46.1 × 2 = 92.2)
Element: Number Sense and Algebra
Sub-element: Interpreting fractions (InF)
InF6 — Fractions as numbers
explains the equivalence of decimals to benchmark fractions (e.g. 1/4 = 0.25, 1/2 = 0.5, 3/4 = 0.75, 1/10 = 0.1, 1/100 = 0.01)
InF7 — Comparing fractions
understands the equivalence relationship between a fraction, decimal and percentage as different representations of the same quantity (e.g.½ = 0.5 = 50% because five is half of ten and fifty is half of 100)
Element: Number Sense and Algebra
Sub-element: Understanding Money (UnM)
UnM6 — Working with money additively
calculates the total cost of several different items in dollars and cents
counts the change required for simple transactions to the nearest five cents
calculates the change, to the nearest five cents, after a purchase using additive strategies
UnM7 — Working with money multiplicatively
connects the multiplicative relationship between dollars and cents to decimal notation (e.g. explains that a quarter of dollar is equal to $0.25 or 25 cents; calculates what 150 copies will cost if they are advertised at 15c a print and expresses this in dollars and cents as $22.50)
solves problems, such as repeated purchases, splitting a bill or calculating monthly subscription fees, using multiplicative strategies
Note, money is not a true decimal system, it is based on the face value of coins and notes rather than a place value system. Understanding how to use currency draws on both additive and multiplicative strategies. Giving change requires being able to round values and work with multiples of 5, 20 or 50 as well as 10, which are the existing Australian coins in operating with money.