Key Ideas for Teaching
Big Ideas
Misunderstandings
Mathematical
Language
Concrete
Resources
Informal Addition
Key Ideas
1. Addition begins informally when students are working their way through the ‘making groups’ stage of learning.
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They learn to combine smaller groups to make larger groups and communicate their learning in words and using sentences.
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At this early stage of their learning, students do not use the operator symbols and =.
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This is because students are still practicing and developing their number sense and these symbols add unnecessary cognitive load.
2. When beginning addition with students, ask them to start with two groups [of counters], they don’t have to be equal, and then combine the groups to make one whole group.
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Prompt students to organise their counters in a way that makes counting (or subitising) more efficient.
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As a scaffold, you may provide students with a tens frame and ask them to place their groups together on the one (or more) tens frame.
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To show the combined groups more explicitly, students could be prompted to collect separate-coloured counters for each group.
3. When describing their additions, ask students to write down their processes using operator words such as, ‘and’, ‘more’, ‘altogether’, ‘makes’.
4. Students are ready for the use of the operator symbols +, -, and = when they have shown proficiency in combining groups and communicating the steps correctly both verbally and in writing.
Informal Subtraction
Key Ideas
1. Subtraction using take-away
a. Students are ready for subtraction by take-away when they are comfortable adding groups of different sizes without group markers.
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When first introducing students to subtraction use the word take-away instead of subtraction.
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The term subtraction can be introduced when proficiency is shown in this area.
b. When first learning subtraction as take-away, a degree of ‘number sense’ needs to be demonstrated by the students.
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Students need to make the connection that they must start with the larger group and then take away the smaller group.
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Allow them to explore this concept themselves using concrete materials instead of pointing it out for them. This will ensure a conceptual understanding of take-away with natural numbers.
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*Note: It is mathematically incorrect to suggest that they can’t take a larger number away from a smaller number (you can, it just makes a negative number).
c. Once students have formed their larger group, they will physically take objects away to show their final amount.
d. When describing their subtraction as take-away, ask students to write down their process initially using the word take-away and then when more confident using different words like ‘less’.
2. Subtraction using difference
a. Students are ready for subtraction by difference when they are comfortable adding groups of different sizes without group markers.
b. Subtraction by difference is a comparison model.
c. Students make groups and compare them to work out ‘how many more’ or ‘how many less’, one group is from another.
d. This value will be the difference of the subtraction and can be worked out using the ‘adding on’ strategy, or by taking away.
It will take time for students to develop the concept of adding and subtracting. Continuous exposure to problems, use of think boards to organise thinking and repeated use of concrete materials will help to support this development.
Additional Activities
Links to external websites:
Counting principles addressed:
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Number sequences
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Skip counting
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Odd and even numbers
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Operations (addition and subtraction)
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Year 1*
recognise, copy and continue repeating patterns represented in different ways
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
*NOTE: This activity can be differentiated to include similar Number and Algebra content descriptors from
F - 6
Counting principles addressed:
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Ordinal numbers
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Sequences
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Operations (addition and subtraction)
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Year 1*
recognise, copy and continue repeating patterns represented in different ways
recognise, continue and create repeating patterns with numbers, symbols, shapes and objects, identifying the repeating unit
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
*NOTE: This activity can be differentiated to include similar Number and Algebra content descriptors from
F - 6.
Counting principles addressed:
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Part-part-whole
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Formal addition
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Formal subtraction
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Year 1
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
Counting principles addressed:
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Recognising numbers
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Formal addition
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Formal subtraction
Content descriptions addressed:
Foundation
AC9MFN01
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
Year 1
AC9M1N04
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
Counting principles addressed:
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Cardinality
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Trusting the count
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Informal addition
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Informal subtraction
Content descriptions addressed:
Foundation
AC9MFN01
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
AC9MFN04
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Year 1
AC9M1N04
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
Counting principles addressed:
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Formal addition
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Formal subtraction
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Number lines
Content descriptions addressed:
Year 1*
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
*NOTE: This activity can be differentiated to include similar Number and Algebra: Addition and Subtraction content descriptors from F - 6
Counting principles addressed:
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Formal addition
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Formal subtraction
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Number lines
Content descriptions addressed:
Foundation
name, represent, and order numbers including zero to at least 20, using physical and virtual materials and numerals
partition and combine collections up to 10 using part-part-whole relationships and subitising to recognise and name the parts
Year 1
add and subtract numbers within 20, using physical and virtual materials, part-part-whole knowledge to 10 and a variety of calculation strategies
Number principles addressed:
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Place Value
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Formal addition
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Formal subtraction
Content descriptions addressed:
Year 1
AC9M1N04
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
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
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: Addition and Subtraction content descriptors from F - 6
References
Reys R., et.al, (2022) CHAPTER 8: Extending number sense: place value, Helping Children Learn Mathematics. John Wiley & Sons Australia, Ltd.
Rogers, A., (2017), Teaching Place Value: A Framework, Prime Number: Volume 32, Number 1, pp 19-21. The Mathematical Association of Victoria.
Rogers, A. (2014). Investigating whole number place value assessment in Years 3-6: Creating an evidence-based Developmental Progression. [Unpublished PhD thesis]. RMIT University.
Siemon, D., Beswick, K., Brady, K., Clark, J., Faragher, R. and Warren, E. (2011) Teaching mathematics: Foundations to middle years, Oxford University Press.
Van de Walle, J., Karp, K., M, B.-W. J. & Brass, A., 2019. Primary and Middle Years Mathematics: Teaching Developmentally. Australia: Pearson
