ADDITION and SUBTRACTION
Friends of 10, Partitioning, Place Value, Compensation, Estimation, Doubling, Halving,
Inverse Operations, Equality, Commutativity, Associativity
Concept Development
When children first begin to add and subtract, they do it by counting by ones. So let's start by
investigating just how much deep understanding there is in counting by ones!
For the next 2 paragraphs, the letters m n o p q r s t u v are the numerals 0 1 2 3 4 5 6 7 8 9
Show me q fingers ... Did you count from n (o is 0)? Now that you know that q is 4, you also know
that q will always be 4 because you have number sense. Children learning to count items do not
initially have number sense - they will have to count to 'q' many times before they develop their
number sense
What is s plus r? Did you say 'x'? Did you forget our number system is base 10 and that after the
tenth numeral, we start to repeat numerals? Would s plus r be 6 plus 5, which is 11, and so be
represented by nn?
Even with number sense, counting and addition are challenging!
Children will initially count items from one. When adding 3 and 4, they will get 4 items, get
another 3 items, and count all items from 1. When subtracting 3 from 5, they will get 5 items, take
away 3 items, and count the remaining items from 1. Recording at this level is informal - the
addition, subtraction and equals signs are not introduced until after children have developed deep
understanding of the concepts of addition, subtraction and equality
As children develop their number sense, they will begin to add and subtract without items, but
will still count from 1. When adding 3 and 4, they will imagine 4 items in their heads, add another
3 items in their heads, and count all items from 1. When subtracting 3 from 5, they will imagine 5
items in their heads, imagine taking away 3 items, and then try to count the remaining items from
1. Because of the difficulty in doing this, children do not generally go through this level for
subtraction. Most will still need items for subtraction while counting from 1 in their head for
addition, then move to the next level for addition and subtraction. Recording at this level is formal
- using addition, subtraction and equals signs in horizontal number sentences
As children further develop their number sense, they will begin to count on by ones from one
number to add, and count back by ones from one number to subtract
All of these levels involve children in counting by ones. Introducing the algorithm while children
are still counting by ones limits their understanding of important mathematical concepts,
properties of the operations of addition and subtraction and the inverse relationship between
addition and subtraction. Indeed the algorithm can be (an often is) done by counting by ones! The
algorithm should not be introduced to children until they are able to add and subtract four-digit
numbers without an algorithm.
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Adding and Subtracting without counting by ones involves using Friends of 10, Partitioning,
Place Value, Compensation, Estimation, Doubling, Halving, Inverse Operations, Equality,
Commutativity, Associativity - all very important concepts, relationships and properties of
number and operations
Let's investigate adding 2 four-digit numbers to see how we use these concepts, relationships and
properties:
How could we partition 5897 to add it to 6825?
Did we partition using place value?
How could we use friends of 10 to add 5 thousand
to 6 thousand 825?
How did we partition 5000?
How could use place value to add the other
thousand?
How could we use friends of 10 to add 8 hundred to
11 825?
Did seeing 1 thousand flexibly as 10 hundred help?
Did we see 2 thousand as 20 hundred?
How did we partition the 800?
How could use place value to add the other 6 hundred?
How could we use friends of 10 to add 9 tens to 12 625?
Did seeing 1 hundred flexibly as 10 tens help? Did we see 7 hundred as 70 tens?
How did we partition 90?
How could we use place value to add the other ten?
How could we use friends of 10 to add 7 ones to 12 715?
How did we partition the 7?
How could we use place value to add the other 2 ones?
Let's investigate adding these numbers using compensation:
Who noticed that 5897 is almost 6 thousand?
How many do we need to add to 5897 get to 6000?
Will100 get us to 5997, then another 3 get us to 6000?
Could we add 6000, then subtracted 103?
What is 6825 plus 6000?
How did we use place value to add 6825 plus 6000?
What is 12 825 minus 103?
Will we have one less hundred? Will we have 3 less ones?
Are you thinking the algorithm is easier? Are you thinking you don't have to think nearly so much
using the algorithm? That's the point! We only think when we have a problem, and the reason we
learn Mathematics is to think!
Do you think you could change these numbers into decimal fractions, for example, 68.25 + 5.897,
or money, for example, $68.25 + $58.97, and add them using concepts, relationships and
properties?
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Email: [email protected] Twitter: @carol_learning Facebook: A Learning Place
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Let's investigate subtracting 2 four-digit numbers to see how we use these concepts, relationships
and properties:
How could we partition 5897 to subtract it from 12
722?
Did we partition using place value?
How could we use place value to subtract 5 thousand
from 12 thousand?
How did we partition the 5000?
How could we use friends of 10 to subtract the other
3000?
How could we use place value to subtract 8 hundreds
from 7 hundreds? Do we actually have 77 hundreds?
How did we partition the 800?
How could we use friends of 10 to subtract the other 100?
How could we use place value to subtract 9 tens from 2 tens? Do we actually have 92 tens?
How did we partition the 90?
How could we use friends of 10 to subtract the other 70?
How could we use place value to subtract 7 ones from 2 ones? Do we actually have 32 ones?
How did we partition the 7?
How could we use friends of 10 to subtract the other 5?
Let's investigate subtracting these numbers using compensation:
Who noticed that 5897 is almost 6 thousand?
How many do we need to add to 5897 get to 6000? Will 100 get us to
5997, then another 3 get us to 6000?
Could we subtract 6000, then add 103?
What is 12 722 minus 6000?
How did we use place value to subtract 6000 from 12 722?
What is 6722 plus 103?
Will we have one more hundred? Will we have 3 more ones?
Are you thinking the algorithm is easier? Are you thinking you don't have to think nearly so much
using the algorithm? That's the point! We only think when we have a problem, and the reason we
learn Mathematics is to think!
Do you think you could change these numbers into decimal fractions, for example, 68.25 - 5.897,
or money, for example, $68.25 - $58.97, and subtract using concepts, relationships and
properties?
Children may be exposed to the algorithms for addition and subtraction in about Year 4, but
continue to add and subtract using concepts, relationships and properties all of their lives!
The Scope and Sequences, Teaching Programs, Teaching Plans, Teaching Videos and Learning
Videos develop the concepts of addition and subtraction and the concepts, relationships and
properties of Friends of 10, Partitioning, Place Value, Compensation, Estimation, Doubling,
Halving, Inverse Operations, Equality, Commutativity and Associativity from K-6
Website: http://www.alearningplace.com.au
Download the App and scan the QR Code
Email: [email protected] Twitter: @carol_learning Facebook: A Learning Place
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