Transcript File - Ballarat Diocese Mathematical Teaching & Learning

Making Subtraction
Concepts Meaningful
Rosemary Reuille Irons
Senior Lecturer
Queensland University of Technology
[email protected] or [email protected]
What is a concept?
A concept is the picture in your mind of an idea.
Images built through language
experiences help develop concepts?
What steps do we follow to develop
operation concepts?
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•
•
•
Child’s Language
Materials Language
Mathematical Language
Symbols
Representations
CONCRETE/
VISUAL
VERBAL
Student Language
oral  written
SYMBOLIC
Eight mice are playing by the cheese.
Two mice run away.
How many mice are playing now?
Representations
CONCRETE/
VISUAL
VERBAL
Student Language
oral  written
Materials Language
oral and written
SYMBOLIC
Concrete/pictorial materials –
take away
take out 2
8
8 cover up 2
Representations
CONCRETE/
VISUAL
VERBAL
Student Language
oral  written
Materials Language
oral and written
Mathematical Language
oral and written
SYMBOLIC
The new mathematical words that are
used with the concept
8
spend
5
leaves
3
8
take
5
is
3
8
subtract
5
equals
3
Representations
CONCRETE/
VISUAL
VERBAL
Student Language
oral  written
Materials Language
oral and written
Mathematical Language
oral and written
Symbolic Language
written
SYMBOLIC
The mathematical abbreviations and
formulae.
8
5
=
3
Teaching the
Subtraction Concept
Subtraction Concept
Finding the missing part.
The missing part could be what is left after a
take away.
The missing part could be how many to
add on.
The missing part could be the difference in
number.
Rosie needs 12 apples. She has picked 7
apples. How many more apples does she
need?
Rosie had 12 apples in a bag. She took
out 7 apples. How many apples are in the
bag now?
Rosie has 12 red apples and 7 green
apples. How many fewer green apples
does she have?
Take Away
Child’s language
Missing addend
Materials language
Mathematical
language
Symbolic language
Child’s language
Difference
Materials language
Mathematical
language
Symbolic language
Child’s language
Materials language
Mathematical
language
Symbolic language
Child's language
Everyday language – take
away
Eight mice are playing? Two mice run
away? How many mice are playing now?
Child's language
Everyday language – missing addend
There are 8 mice altogether. How many
mice are hiding in the cheese?
Four cars in the carpark. How many more
will drive in to make ten cars in the carpark?
Child's language
Everyday language – difference
Eight mice are playing in front of the cheese.
Two mice are playing in the back. How many
more mice are playing in front?
Materials language
Concrete/pictorial materials –
take away
spend 2
Materials language
Concrete/pictorial materials –
take away
take out 2
8
8 cover up 2
Materials language
Concrete/pictorial materials –
missing addend
There are 8 altogether.
How many are covered?
Materials language
Concrete/pictorial materials –
difference
How much more
is 8 than 2?
8 cover up 2
Make the
number of
objects to
represent
the two
groups.
Cover the
parts of the
groups that
are the same
to show the
difference.
Mathematical language
The new mathematical words that are
used with the concept
subtract
[Try to avoid using the word minus. In
mathematics this is best associated with
negative numbers.]
Symbol language
The mathematical abbreviations and
formulae.
8
5
=
3
What are the features of the stories that
make them all subtraction?
Rosie needs 12 apples. She has picked 7 apples. How many more
apples does she need?
Rosie had 12 apples in a bag. She took out 7 apples. How many
apples are in the bag now?
Rosie has 12 red apples and 7 green apples. How many fewer green
apples does she have?
For each subtraction situation, the total
and number in one part of the total are
known. The unknown value is the other
part of the total.
For addition, 2 or more parts are known.
The unknown value is the total.
Stories provide the opportunity to relate the
operations.
Make sure that both are introduced when
the addition concept is developed.
Relate subtraction to addition
How can you work out the number of
covered dots?
6
13 altogether
Build links to addition during the
work with missing addend
subtraction.
5
8
=
+
5
=
8
Teaching the
number fact
strategies
The approach to number facts
Number facts are best
learned in clusters.
Each cluster is organised
around one strategy – a
strategy that can be
used to learn facts and
then with numbers
beyond the facts.
The stages for each cluster
• introduce the strategy
• reinforce the strategy
• practice the facts
• extend to examples beyond the fact range.
Cluster 1: Count on
Count on 1
Count on 2
and for some students,
Count on 3
6
Cluster 2: Use Doubles
Double
Double-add-1
Double-add-2
Cluster 3: Make Ten
Number facts in this
cluster have one
addend close to 10.
9 + 4 = ____
is the same as
10 + 3 = ____
Teaching the
subtraction number
facts
Use the sequence for addition facts to plan
the sequence for subtraction facts
Count on facts
Use doubles facts
Make to 10 or bridge to 10 facts
For each subtraction cluster, encourage
students to use the strategy ‘think addition.’
The connection between addition and
subtraction is essential.
Begin the links to subtraction when the
addition concept is taught.
The stages for each cluster
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•
•
•
introduce the strategy
reinforce the strategy
practice the facts
extend to examples beyond the fact range.
Introduce the strategy
There were 8 cubes in the cup.
I have taken out 2 cubes.
How many cubes are still in
the cup? What are all of the
ways you know?
8
Count on/Count back subtraction
facts
How can you work out the number of
covered dots?
6
8 altogether
The addition facts
6 + 2 = ___
2 + 6 = ___
are in the count-on cluster.
The related subtraction facts are
8 – 2 = ___
8 – 6 = ___.
Initially, students might work out 8 – 2 =__
using a count back strategy.
Reinforce the strategy
Ask questions such as:
How would you work out the answer?
11 - 9 = ___
How could you work out the number that is
covered?
2 +
= 10
Use addition to plan the sequence
Count on facts
6 + 2 = __
Doubles facts
6 + 7 = __
8 – 2 = __
8 – 6 = __
13 – 6 = __
13 – 7 = __
Use doubles subtraction facts
How can you work out the number of
covered dots?
6
13 altogether
Use addition to plan the sequence
Count on facts
9 + 2 = __
Doubles facts
6 + 8 = __
Make to ten facts
6 + 9 = __
11 – 2 = __
11 – 9 = __
14 – 6 = __
14 – 8 = __
15 – 9 = __
15 – 6 = __
Make to 10 subtraction facts
How can you work out the number of
covered dots?
6
15 altogether
Consideration of interests does
not mean indulging children or
abdicating responsibility. It means
that children are more likely to
find curriculum meaningful and
engaging when it relates to and
respects their interests.
NAEYC- Developmentally
Appropriate Practice 1997
Learning never ends and as
teachers we should approach
each day –
the same way as a child does
everything is a new discovery.
Discover something new each
day about each child in your
learning environment.