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Subtraction and computation
Dr. Calvin J Irons
When/where do we need to be
able to subtract?
What types of situations lend
themselves to subtraction?
I have $125.
If I buy the $89 ticket,
how much money will
I have left?
What pictures do students have of
numbers/operations?
What pictures do students need to
calculate mentally with subtraction?
The mathematical structure of numbers
include:
Counting – a discrete representation:
ungrouped
grouped
place value
Measurement – a continuous representation:
linear
area
Counting
Counting is not an efficient strategy.
The students do (and we stress) too much
counting.
Find the difference
$38
$54
How do we calculate if our picture of
numbers is one that relates more to counting
than some of the other representations?
Place value
Place value is an abstract system that
enables us to read and write numbers.
Find the difference
$38
$54
How do you calculate if the main picture of
number is place value?
Place value
Remember, place value is just one of the
structures. In some situations other
representations may be more useful –
particularly for mental strategies involving
addition and subtraction (and of course for
fractions and even decimals).
Linear
The only model that can be used with ALL
numbers.
The best model for many strategies
Not enough time (probably no time) is
spent to develop this model.
And then, the true linear aspects of the
model are not stressed.
How does a number line show a number?
What is 38 on this number line?
0
10
20
30
40
50
60
70
80
90
100 110 120
When asked, most people say a number such
as three is at a particular ‘point.’
A number is a length (starting at 0).
When students are asked to work with a
number such as thirty-eight, what ‘picture’
do they use?
How would you work out the
difference between the prices of
these two items?
$53
$38
Some important and less structured
features of numbers include being able
to :
Partition a number in multiple ways:
Double/halve with confidence:
Are
Aretheir
theirother
otherpictures
picturesthat
thatwe
weuse
usewhich
to are not
based on a strict mathematical structure?
Find the difference? How could you think?
$160
$190
How could we work out the difference
in price of these two items?
$3.75
$1.75
$1.95
$1.95
Addition Strategies
Begin by extending a fact strategy
First
Further
decimal
Strategies extension extensions extensions
Count-on
6+1
9+2
Count-on
16 + 1
19 + 2
Count-on
26 + 21
29 + 12
Count-on
3.6 + 2.1
2.9 + 1.2
Use doubles
7+7
6+5
Use doubles
25 + 25
26 + 25
Use doubles
27 + 27
126 + 125
Use doubles
2.5 + 2.5
1.26 + 1.25
Bridge-ten
9+4
Bridge-ten
39 + 4
Bridge-ten
198 + 25
Bridge-ten
1.98 + 0.6
Subtraction Strategies
Begin by extending a fact strategy
First
Further
Strategies
extension
extensions
Take small
6–1
9–2
Take small
16 – 1
59 – 21
Use addition
6–5
9–7
12 – 6
15 – 7
14 – 9
Use addition
26 – 25
19 – 17
120 – 60
30 – 15
23 – 9
Use addition
no bridging
67 – 53
bridging
85 – 59
126 – 98
What influences the teaching sequence
for subtraction?
The number combinations involved?
The subtraction situations?
The type of problem and choice of
numbers influence how you think.
Change either, and you may want to
take-away subtraction
use a different strategy.
Joel has $75 for the day at Maze World.
How much does he have left after paying
the $29
$24 admission charge?
What is the difference between the prices
of these computer games?
difference subtraction
$75
$29
How much more do you need?
$35
missing addend subtraction
How much more do you need?
$8.35
missing addend subtraction
Bridging: Subtraction facts
Use one board between 2 players. Roll both cubes.
Write the numbers with the answer. Get three in a line.
Cube A: 8, 8, 8, 9, 9, 9
Cube B:11,12,13,14,15,16
2
2
3
3
3
4
4
5
5
5
6
6
6
7
7
7
8
8
Bridging: beyond subtraction facts
Cube A: 8, 18, 28, 9, 19, 29
Cube B:31,32,33,34,35,36
Use one board between 2 players. Roll both cubes.
Write the numbers with the answer. Get three in a line.
2
3
4
5
6
7
8
12
13
14
15
16
17
18
22
23
24
25
26
27
28
Bridging: subtracting decimals
Cube A: 1.8, 2.8, 3.8, 1.9, 2.9, 3.9
Cube B: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6
0.2 0.3 0.4 0.5 0.6 0.7 0.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
2.2 2.3 2.4 2.5 2.6 2.7 2.8
0.2 0.3 0.4 0.5 0.6 0.7 0.8
1.2 1.3 1.4 1.5 1.6 1.7 1.8
2.2 2.3 2.4 2.5 2.6 2.7 2.8
Mental strategies for subtraction
$160
1. Use addition
$190
Mental strategies for subtraction
$160
2. Use place value (in some way)
$190
Mental strategies for subtraction
$160
$190
3. Use partitioning (other than place value) –
and possibly use another strategy
Where is subtraction used outside the
realm of whole numbers and decimals?
0
1
2
0
1
2