Key Ideas for Teaching
Big Ideas
Misunderstandings
Mathematical
Language
Concrete
Resources
Informal Place Value
Unitising
Key Ideas
Students first begin to learn place value informally. This means working with numbers from 1 – 20 and building their part-part-whole knowledge of these numbers. Tens frames are a very effective tool to help students to consolidate their understanding of numbers to ten (friends of 10), and practise unitising once teen numbers are introduced. This is because tens frames allow students to subitise in groups of five and eventually be able to recognise the unit of one ten.
Numbers beyond 10
Students often have trouble partitioning and writing their numbers beyond 10 because of the inconsistency with our number naming system. The ‘teen’ numbers and the ‘ty’ numbers are not only written differently but the teen numbers have some different number naming conventions with 11 and 12 and even 13 and 15. There have been many suggestions for ways around this but sometimes understanding the etymology of a word allows some clarity for both teachers and students.
The word ‘teen’ came from Old English and means ’10 more than’, so the number sixteen means ten more than six, or ‘6 and 10’. This concept can be practised with the repeated use of tens frames and representing the teen numbers in different ways using the number triad.
Start with 14, 16, 17, 18, 19 because these numbers say the digit name in the number. Then introduce practice with 15, 13, 12 and 11 as these are more abstract.
Activity Ideas
*Videos
Place Value principles addressed:
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Informal place value
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Unitising
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Teen numbers
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Adding and subtracting numbers
*Content descriptions addressed:
AC9MFN01, AC9MFN04, AC9M1N01, AC9M1N04, AC9M2N04
*See Instructions for full details
Place Value principles addressed:
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Informal place value
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Factor-10 counting
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Numbers beyond 20
*Content descriptions addressed:
recognise, represent and order numbers to at least 120 using physical and virtual materials, numerals, number lines and charts
quantify sets of objects, to at least 120, by partitioning collections into equal groups using number knowledge and skip counting
Using Models in Place Value
Key Ideas
There are 2 types of models that need to be used when teaching formal place value. These are proportional models and non-proportional models. It is important that students first work predominantly with proportional models in formal place value and then the non-proportional models can be introduced once proficiency has been demonstrated. This is because when students are first learning their grouping by 10, they need to practice perceiving the whole collection before they can move confidently into trading ones for tens (Van de Walle, 2019, p.230).
Proportional Models
These are materials that show the whole quantity (perceptual). *See Concrete Resources for a list of materials.
Non-proportional Models
These are materials where structure or colour denotes value (figurative). *See Concrete Resources for a list of materials.
The Place Value Framework
Key Ideas
The six aspects of the place value framework have been adapted from the work of Dr. Angela Rogers and are interchangeable when teaching place value. For students to be successful in place value and develop a deep conceptual understanding, they need repeated, interleaved practice in all five areas.
1. Make, Name, Represent
When working with place value, provide opportunities for students to make their number using materials, write their number in symbols and words and record their number on their place value charts. Using the number triad board can help students to document their work in an organised way and show it clearly in all of the different ways
2. Count
Whenever students are working with their place value, it is important to provide opportunities for them to practise counting forward and backwards in place value parts. For example, if working with two-digit numbers, counting forwards and backwards in groups of 10 both on and of the decade while using language such as more, less, greater, before, and after.
3. Compare/order
Another important component of learning place value is comparing numbers to determine which is larger and which is smaller. Students can compare numbers by placing them in ascending or descending order, using the symbols <, > and =, or by placing them on an open number line. Number lines are particularly important because the skills learned through number lines can be transferred to many different strands of mathematics. Eventually, students’ comparison of numbers will move from an additive to a multiplicative model. For example, while they may start by comparing 45 and 54 by stating that 54 is 9 more than 45, they will eventually move to comparing 45 to 450 by stating that 450 is ten times greater than 45.
4. Rename
Renaming numbers, also called regrouping, is one of the most important skills that students need to practice when learning place value. It involves partitioning numbers into standard and non-standard place value parts as well as non-place value parts. For example the number 235 can be renamed in standard place value parts as 2 hundreds, 3 tens and 5 ones, non-standard place value parts as 23 tens and 5 ones, or non-place value parts as 1 hundred, 13 tens and 5 ones. Practise in partitioning into non-place value parts is particularly important for developing strong number sense in students.
5. Calculate
When any type of operation is applied to numbers, students need to use their knowledge of place value to complete the calculation successfully. The understanding of the place value system and students’ ability to rename, is crucial for developing the conceptual understanding necessary to complete calculations flexibly. Understanding the place value system also underpins students’ development of number sense which allows for efficiency when working with numbers.
Formal Place Value
Using a place value chart
Key Ideas
The idea of place value refers to the ability to ‘unitise’ which means that one can recognise that ‘10 ones is the same as 1 ten’. Additive place value is the idea that all numbers are composed of additive parts and can be partitioned in both standard place value and non-standard place value. When students are first introduced to place value it is always additively as they practice forming two-digit numbers first and then move on to three-, four-, five-digit, and beyond.
Students can be formally introduced to place value as a system of recording numbers when they:
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Can comfortably count to beyond 20.
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Have a deep understanding of numbers from 1 – 20 including part-part-whole relationships.
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Can trust the count.
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Can interpret and visualise numbers beyond ten in terms of their parts e.g., 1 ten and 3 more is 13.
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Recognise numbers to 10 as countable units.
Learning place value formally involves using a place value chart. Students begin by making 2-digit numbers using proportional models such as bundling sticks or unifix.
1. Introducing two-digit numbers
When reading a place value chart, students first need practice writing their numbers in standard place value parts. Up until this point, students have been learning place value informally with teens and tens frames and the introduction of the place value chart would begin with their teens as well.
The Trading Game is a great way to help students to understand how to build their formal place value knowledge from ‘teen’ to ‘ty’ numbers. In figure 1 the number sixteen is represented on the place value chart using unifix. Figure 2 shows how students would read the chart to write and represent their numbers in standard place value parts.
Figure 1
Figure 2
As students continue to play the trading game their numbers will increase beyond 20. At this stage in their learning, students can now be introduced to writing their numbers in both standard and non-standard place value parts. Figure 3 shows the number 26 written in standard and non-standard place value parts.
Figure 3
Writing numbers in these different ways is called renaming or regrouping. Renaming is a fundamental part of place value and practice in this area is crucial for the development of conceptual place value, which in turn supports the successful development of number sense.
As numbers increase to beyond 30, students may start to recognise a pattern of ‘digit’ then ‘ty’ then ‘ones’. Professor Dianne Siemon suggests that to develop a deep understanding of place value of numbers from 20 – 99, “start with numbers that are generally supportive of the pattern”, (Siemon 2011, p. 299). For example, start with six-ty, seven-ty,
eight-y and nine-ty because these sound more like a count of ones and tens than for-ty, fif-ty, thir-ty and twen-ty.
Activity Ideas
*Videos
*Note: Any of the activities used in 2-digit and three-digit numbers can be adapted to suit numbers beyond 1000.
Place Value principles addressed:
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Figurative place value of two-digit numbers
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Representing numbers using materials
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Renaming using standard place value
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Renaming using non-standard place value
Content descriptions addressed:
recognise, represent and order numbers to at least 120 using physical and virtual materials, numerals, number lines and charts
partition one- and two-digit numbers in different ways using physical and virtual materials, including partitioning two-digit numbers into tens and ones
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 6.
Place Value principles addressed:
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Factor-10 counting – forwards and backwards
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Counting across the decade
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Counting across the century
Content descriptions addressed:
quantify sets of objects, to at least 120, by partitioning collections into equal groups using number knowledge and skip counting
recognise, continue and create pattern sequences, with numbers, symbols, shapes and objects, formed by skip counting, initially by twos, fives and tens
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 6.
Place Value principles addressed:
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Ordering numbers on a number line
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Comparing numbers
Content descriptions addressed:
recognise, represent and order numbers to at least 120 using physical and virtual materials, numerals, number lines and charts
partition one- and two-digit numbers in different ways using physical and virtual materials, including partitioning two-digit numbers into tens and ones
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 6.
Place Value principles addressed:
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Making numbers using a non-proportional model
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Renaming using standard place value
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Renaming using non-standard place value
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Ordering numbers on a number line
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Comparing numbers
Content descriptions addressed:
AC9M1N03
quantify sets of objects, to at least 120, by partitioning collections into equal groups using number knowledge and skip counting
AC9M1A01
recognise, continue and create pattern sequences, with numbers, symbols, shapes and objects, formed by skip counting, initially by twos, fives and tens
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 6.
2. Introducing three-digit numbers
The trading game can continue to be used when moving students onto three-digit numbers. This will emphasise the ’10 of these is 1 of these’ construct, as students begin to bundle their 10 tens to make 1 hundred.
The concept of renaming becomes particularly important here as the ways that numbers can be regrouped into non-standard place value parts increases. See figure 4.
Figure 4
As students continue to make, name and record three-digit numbers, ensure that they are also, regularly practicing:
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The six elements of their Place Value Framework. (*See Part 4)
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Making, naming, and recording three-digit numbers containing teens, e.g., 411, 817, 912.
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Making, naming, and recording three-digit numbers containing internal zeros, e.g., 305, 800, 109.
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Renaming in non-place value parts. These are parts that don’t necessarily include the original digits in the number, e.g., 116 can be renamed as 9 tens and 26 ones.
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Writing the number in expanded form in both standard and non-standard parts, e.g., 116 can be written 100 + 10 + 6, or 110 + 6, or 90 + 26.
During work with three-digit numbers, students should have had extensive practice working with non-proportional models such as MAB, unifix and money. Money is an excellent example of a non-proportional model. It provides students with a way of renaming quantities into ones ($1), tens ($10) and hundreds ($100), as well as into other values that are represented by different notes and coins. For example, $127 can be regrouped into $100 + $20 + $7, or $50 + $50 + $10 + $10 + $5 + $2.
3. Introducing numbers beyond 1000
After repeated practise working with three-digit numbers, students are ready to be introduced to four-digit numbers. Once students can read a three-digit number, they possess the knowledge to read numbers up to millions and beyond (Rogers, 2017). This is because our numeration system is comprised of a pattern of ones, tens, and hundreds in all numbers beyond 1000.
When students are ready to begin work with four-digit numbers, they should be introduced to the new unit of 1000 in the same way as the new unit of 10. That is ‘1000 of these is one of those’ (Siemon, 2011). This notion involves the re-use of the place value pattern of ones, tens, and hundreds to now count thousands and beyond. For example, the number 78 423 is said seventy-eight thousand, four hundred and twenty-three and not seven ten thousands, eight thousands, four hundred and twenty-three. This HTO pattern is referred to as the ‘new place value pattern’ because it repeats as the number of digits in our numbers continues to increase beyond hundreds of thousands and millions.
Figure 5
Figure 5 shows how the new place value pattern repeats for numbers beyond thousands. The numbers represented on this chart are said and written:
73 thousand 289
315 thousand 615
2 million 703 thousand 703
Activities
*Note: Any of the activities used in 2-digit and three-digit numbers can be adapted to suit numbers beyond 1000.
Race to 100 000
Place Value principles addressed:
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Figurative place value of numbers beyond 10 000
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Renaming using standard place value
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Renaming using non-standard place value
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Adding and subtracting using place value parts
Content descriptions addressed:
Year 3
Year 4
Estimating and Rounding
Key Ideas
Estimation is an important skill in mathematics because it allows us to be able to judge the reasonableness of an answer. When performing calculations, estimation means that we can check whether our answers make sense or are within a reasonable range. Estimation makes up a large part of our daily lives. We are more likely to estimate when working with things like money or in cooking than we are to calculate the exact amount. Rounding makes up a large part of our ability to estimate quantities.
Multiplicative Place Value
Key Ideas
Multiplicative place value involves a formal understanding of the patterns in our base-10 number system. It focuses on the idea that multiplying by 10 causes the digits in the number to move one place to the left and dividing by 10 moves them one place to the right. Understanding the multiplicative nature of place value allows students to draw on efficient strategies when multiplying or dividing numbers. This includes using:
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The commutative property.
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The associative property.
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The distributive property.
Multiplicative place value is a precursor to a deeper understanding of decimals as decimal fractions.
Multiplicative Place Value
Bridging to decimals
Key Ideas
By the end of year 3 students should have spent time working with problems that allow them to investigate the multiplicative nature of our base-10 system. Once proficiency is attained in this area, they can be introduced to decimals as decimal fractions. Before introducing the decimal point, it is important to allow students to make the connection to multiplying by a power of 10 to get a larger number and dividing by a power of 10 to make an increasingly smaller base-10 fraction.
Additional Activities
Links to external websites:
Place Value principles addressed:
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Making numbers
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Ordering numbers
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Comparing numbers
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Numbers on an open number line
Content descriptions addressed:
Year 1
recognise, represent, and order numbers to at least 120 using physical and virtual materials, numerals, number lines, and charts
Year 2
recognise, represent, and order numbers to at least 1000 using physical and virtual materials, numerals, and number lines
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 8.
Place value principles addressed:
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Making numbers
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Comparing numbers
Content descriptions addressed:
Year 2
recognise, represent, and order numbers to at least 1000 using physical and virtual materials, numerals, and number lines
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 8.
Place value principles addressed:
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Making numbers
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Ordering numbers
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Comparing numbers
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Numbers on an open number line
Content descriptions addressed:
Year 1
recognise, represent, and order numbers to at least 120 using physical and virtual materials, numerals, number lines, and charts
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 8.
Place value principles addressed:
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Making numbers
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Comparing numbers
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Adding multi-digit numbers
Content descriptions addressed:
Year 1
recognise, represent, and order numbers to at least 120 using physical and virtual materials, numerals, number lines, and charts
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10, and a variety of calculation strategies
Year 2
recognise, represent, and order numbers to at least 1000 using physical and virtual materials, numerals, and number lines
add and subtract one- and two-digit numbers, representing problems using number sentences, and solve using part-part-whole reasoning and a variety of calculation strategies
Year 3
recognise, represent, and order natural numbers using naming and writing conventions for numerals beyond
10 000
add and subtract two- and three-digit numbers using place value to partition, rearrange, and regroup numbers to assist in calculations without a calculator
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 8.
Place value principles addressed:
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Making numbers using a non-proportional model
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Renaming using standard place value
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Renaming using non-standard place value
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Order of operations
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Calculating
Content descriptions addressed:
Year 1
recognise, represent, and order numbers to at least 120 using physical and virtual materials, numerals, number lines, and charts
partition one- and two-digit numbers in different ways using physical and virtual materials, including partitioning two-digit numbers into tens and ones
Year 2
recognise, represent, and order numbers to at least 1000 using physical and virtual materials, numerals, and number lines
partition, rearrange, regroup and rename two- and three-digit numbers using standard and non-standard groupings; recognise the role of a zero digit in place value notation
Year 3
recognise, represent, and order natural numbers using naming and writing conventions for numerals beyond
10 000
add and subtract two- and three-digit numbers using place value to partition, rearrange and regroup numbers to assist in calculations without a calculator
*NOTE: This activity can be differentiated to include similar Number and Algebra: Place Value content descriptors from 1 - 8.
References
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Reys R., et.al, (2022) CHAPTER 8: Extending number sense: place value, Helping Children Learn Mathematics. John Wiley & Sons Australia, Ltd.
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Rogers, A., (2017), Teaching Place Value: A Framework, Prime Number: Volume 32, Number 1, pp 19-21. The Mathematical Association of Victoria.
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Rogers, A. (2014). Investigating whole number place value assessment in Years 3-6: Creating an evidence-based Developmental Progression. [Unpublished PhD thesis]. RMIT University.
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Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R. and Warren, E. (2011) Teaching mathematics: Foundations to middle years, Oxford University Press.
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Van de Walle, J., Karp, K., M, B.-W. J. & Brass, A., 2019. Primary and Middle Years Mathematics: Teaching Developmentally. Australia: Pearson
