# National Numeracy Learning Progression

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## Operating with decimals description

This sub-element focuses on understanding the use of place value in operating with decimals. Decimals are better suited to estimating magnitude than fractions because decimals use the base-ten system to record quantity and fractions do not. However, the base-ten system used with whole numbers can also contribute to misconceptions with decimals. For example, recognising that whole numbers with more digits are always larger and applying this to decimals may lead to incorrectly believing 0.75 is larger than 0.8 and 0.320 is larger than 0.32. Understanding that fractions with larger denominators result in smaller magnitudes and longer decimals contain smaller parts can lead to believing longer decimals must be smaller than shorter decimals.

Decimals are commonly used to record metric quantities and have applications in areas that range from nutritional advice to expressing tolerances in precision engineering.

(NB: The notation for fractions is distinct from the place value notation used with decimals. This progression treats the development of decimal notation separately from the development of common fractions).

Some students will communicate using augmentative and alternative communication strategies to demonstrate their numeracy skills. This may include digital technologies, sign language, braille, real objects, photographs and pictographs.

Each sub-element level has been identified by upper-case initials and in some cases lower-case letters of the sub-element name followed by ascending numbers. The abbreviation for this sub-element is OwD. The listing of indicators within each level is non-hierarchical. Subheadings have been included to group related indicators. Where appropriate, examples have been provided in brackets following an indicator.

### OwD1

Understanding positional value of decimals

• uses knowledge of positional value of numbers to add and subtract decimals of up to three decimal places

### OwD2

Understanding and estimating relative size of decimals

• interprets the relative size of decimals, and rounds to estimate answers
• estimates the size of answers without doing the exact calculations (1.23 + 3.4 cannot be 1.57 because the sum must be greater than 4)

### OwD3

Understanding the effects of multiplication and division with decimals

• understands that multiplying and dividing decimals by 10, 100, 1000 changes the positional value of the numerals
• explains that multiplication does not always make the answer larger (when multiplying whole numbers by a decimal less than 1, 15 x 0.5 = 7.5)
• connects and converts decimals to fractions to assist in mental computation involving multiplication (to find 16 x 0.25, recognises 0.25 as a quarter, and finds a quarter of 16)
• connects and converts decimals to fractions to assist in mental computation involving division (to determine 0.5 ÷ 0.25, recognises the answer is 2 as there are two quarters in one-half)
• recognises the equivalence of decimals to benchmark fractions ( $$\frac14$$ = 0.25, $$\frac12$$ = 0.5, $$\frac34$$ = 0.75, $$\frac1{10}$$ = 0.1, $$\frac1{100}$$ = 0.01)

### OwD4

Flexible strategies for multiplication and division of decimals

• uses knowledge of positional value of numbers to multiply and divide decimals
• uses knowledge of approximate answers to check accuracy of solutions when using a variety of strategies