ADDITION AND SUBTRACTION STRATEGIES
Strategies are flexible. Students should not be given a prescribed sequence of
steps to follow when using a particular strategy. It is likely to see each of the
following strategies used in a variety of ways during a Number Talk or math
lesson. The examples, models, and recording schemes given are not a
comprehensive list of all ways in which the strategy can be used, rather, a small
sample to provide illustrations of the strategy. Allow students to use the method
that works for them. Students will eventually begin to perform these
computations mentally without the need to record or model. This is the goal and
should be encouraged. Note: Students should not be expected to know the strategy by
name. Each strategy below is referred to by several different names. Teachers may refer to a
particular strategy as “Peter’s strategy” as if Peter was the first to present that strategy.
Two common components of strategies for mental addition and subtraction are decomposing
one or more of the addends and compensation.
Decomposing
Decomposing (sometimes called splitting) is the practice of breaking numbers apart.
Numbers can be decomposed in many way. Students should become flexible with
decomposing in order to select appropriate combinations that make sense to the
individual and are efficient for the given problem.
Example: Some of the ways 78 can be decomposed
75 + 3
70 + 8
20 + 50 + 8
72 + 6
25 + 50 + 3
60 + 18
Compensation
Compensation is the process of adjusting one or more of the numbers in a problem prior
to performing the computation and later readjusting in order to compensate. When
adding 25 + 27 a student may adjust the 27 to make the problem 25 + 25 = 50, and
then compensate for the adjustment by adding 2, 50 + 2 = 52. A student may also
adjust the 25 and the 27 to make the problem 22 + 30. In this case, 3 was added to the
27, making 30, and then to compensate the 3 was taken from the 25.
Base ten blocks, open number lines, hundred charts, and drawing are all good models
for students to use when learning strategies for mental math.
A closed number line is a line with prerecorded tick
marks. It is often used when students are computing
smaller numbers and for learning strategies with
single- and sometimes double-digit addition and
subtraction. Students’ first exposure to number lines
should be with closed number lines and jumps of one.
An open number line is customized by creating
tick marks as a problem is solved. It is used
when students have a basic understanding of
making jumps on closed number lines. It is
primarily used when larger numbers or larger
jumps need to be made.
EXAMPLES OF ADDITION STRATEGIES
CHUNKING (ALSO REFERED TO AS JUMPS)
Early use of this strategy involves counting on one at a time and evolves into counting
on in jumps that make sense to the individual.
Example: 26 + 47
Students may choose to add the tens and then add the ones. Some students
will add each individual ten and others may see a more efficient “jump” of 40.
Students may jump the 7 one at a time or may jump to the next benchmark
number and then jump the rest.
Possible Models
Open Number Line
26 + four jumps of ten (40) + seven
jumps of one (7)
26 + one jump of forty (40) + seven
jumps of one (7)
+ 40
+ 40
+10 +10 +10 +10 +1 +1 +1 +1 +1 +1 +1



26 36 46 56 66 67 68 69 70 71 72 73
26 + one jump of forty (40) + one jump
of four (4) + one jump of three (3)
+ 40
26
(26) add the tens (26 + 40)
then add the ones (66 + 7)
  26

66 70 73
and compose a ten with ten of the ones


Hundred Chart
2
12
3
13
4
14
5
15
21
31
22
32
23
33
24
34
25
35
41
51
42
52
43
53
44
54
45
55
61
71
62
72
63
73
64
74
65
75
66 67 68 69 70 71 72 73
Base Ten Blocks
+4 +3
1
11
+1 +1 +1 +1 +1 +1 +1
+1 +1 +1 +1 +1 +1 +1
6
16
26
36
7
17
8
18
9
19
10
20
27
39
28
38
29
39
30
40
46
56
66
76
47
57
48
58
49
59
50
60
67
77
68
78
69
79
70
80
Move down four (+ 40), then right 7
(+7) which requires wrapping to the
next line.
Students may also choose to add ones to get to the next benchmark, add tens, and
then add the remaining ones.
Example: 26 + 47
Add 4 to get to 30, then add 40 to get to 70, and then add the remaining 3 to get to
73.
Possible Models
Number Line
+4
+40
+3


26 30
70 73
Possible Recording Schemes
+4 +10 +10 +10 +10 +3

26 30
40
50
60
26 + 40 = 66
66 + 4 = 70
70 + 3 = 73
26
30
+ 4 + 40
30
70
70
+ 3
73
70 73
26 + 4 + 10 + 10 + 10 + 10 + 3 = 73
EXAMPLES OF ADDITION STRATEGIES
ANCHOR TO TENS
This strategy supports place value understanding and number sense. This strategy
involves decomposing one of the addends to make the other addend a multiple of ten.
In later grades, this strategy is expanded to “Make 100” or “Make 1000”, etc.
Example 1: 28 + 7
Students can add 2 to 28 to make the next ten, or 30, and then add the remaining
5 for an answer of 35.
Example 2: 46 + 38
Students can take 2 from the 46 and add to the 38 to make 44 + 40 which is 84.
OR
Students can take 4 from the 38 and add to the 46 to make 50 + 34 which is 84.
Possible models
Base Ten Blocks
28
+ 7
       
Hundred Chart
Open Number Line
28 + 7
+2 to the
next “10”
+5 for a
total of 7
Becomes 30 + 5
        

28
30
35
1
11
21
31
2
12
22
32
3
13
23
33
4
14
24
34
5
15
25
35
6
16
26
36
7
17
27
39
8
18
28
38
28 + 2 + 5
30 + 5 = 35
28 + 7 =
28 + 2 + 5=
30 + 5 = 35
10
20
30
40
Begin at 28, move 2 to next “10” and the 5
more for total of 7.
In Example 2, begin at 46 move 4 to next “10”
then down 3 (+30) to 80 then move the
remaining 4 to 84
Possible recording schemes
28 + 7
9
19
29
39
28 + 7
2 + 5
30 + 5 = 35
MAKING EASY NUMBERS
Example: 50 + 78
Students may recognize 50 as an easy number to work with and decompose the
78; 50 + 50 + 28 = 100 + 28 = 128
OR
Students may find it easy to add 50 + 75 and then 3 more by to decomposing 78
into 75 + 3; 50 + 75 + 3 = 125 + 3
Example: 27 + 52
Students may recognize 25 and 50 as easy number to work with and decompose
27 into 2 + 25 and 52 into 50 + 2; 2 + 25 + 50 + 2 = 2 + 75 + 2 = 79.
This strategy is very flexible and can be used in many ways to solve problems.
Different people will find different numbers easier to work with. Students should be
encouraged to use numbers that they find make the problem simpler for them.
DOUBLES
This strategy is useful when adding two numbers that are close to each other.
Example: 45 + 48
Students can double 45 to get 90 and then add the remaining 3 to get 93.
EXAMPLES OF ADDITION STRATEGIES
PARTIAL SUMS (ALSO REFERRED TO AS EXPANDED FORM)
Expanding is a form of decomposing. Students can use their place value understanding
to add or subtract large numbers.
Example: 236 + 183
(200 + 30 + 6) + (100 + 80 + 3) = 300 +110 + 9 = 419
Possible Models
Base Ten Blocks
236
+
Possible Recording Schemes
Right to Left
Left to Right
236
236
+183
+183
9 (add ones)
300 (add hundreds)
110 (add tens)
110 (add tens)
300 (add hundreds)
9 (add ones)
419 (add the sums)
419 (add the sums)
183
       
Combine the hundreds, the tens,
and the ones

Compose a hundred with 10 of
the tens
236 200 + 30 + 6
+183 +100 + 80 + 3
300 +110 +9 = 419
 = 419
COMPENSTATION
Compensation is the ability to add or subtract from a number and then compensate for
the difference when solving.
Note: Students may refer to it as “easy”, “benchmark”, “friendly” or “anchor” numbers.
Example: 56 + 38
Students may change 38 to 40, add 56 + 40 to get 96 and then subtract the 2
they had added, to get 94.
OR
Students may add 4 to the 56 and 2 to the 38, changing the problem to 60 +
40 = 100 and then subtract the 2 and 4 (or 6); 100 - 6 = 94.
OR
Students may add 2 to the 38 and compensate by subtracting 2 from the 56,
changing the problem to 54 + 40 = 94.
Possible Models
Possible Recording Schemes
Base Ten Blocks
Number Line
+40
-2

56

94 96
56 add 40 and
then compensate
by moving back 2
56
+
38
+  
Becomes 56 + 40 by adding 2
to 38
+  =
Compensate by
removing 2
    (94)
56 + 38
+2
56 + 40 = 96
-2
94
56 + 38+2
56 + 40
96-2
94
56 + 38 + 2 = 96
96 – 2 = 40
EXAMPLES OF SUBTRACTION STRATEGIES
Real-Life Context
Providing students with real-life context gives meaning to the numbers and the
operation. Students will more readily see the connection between addition and
subtraction and understand the various meanings of subtraction, take from/add to, take
apart/put together, and compare when problems are presented in context. Presenting
students will all problem types prevents the common misconception that the definition of
subtraction is “take away”.
ADDING UP
This strategy reinforces the connection between addition and subtraction. This strategy
is similar to the addition strategy ADDING UP IN CHUNKS. Problems involving “How
many more?” promote the use of this strategy. Students begin by counting up one at a
time and progress to counting up in chunks that make sense to them.
Example: “Sarah’s goal for the walkathon is to walk 82 laps. She has walked 26
laps so far. How many more does she need to walk?” 26 + ___ = 82, or 82 – 26
Students may determine the number of jumps to get to the next benchmark
number, 26 + 4 = 30. This may be followed by adding up in chunks of ten, 30 +10
= 40, +10 more = 50, +10 more = 60, +10 more = 70, +10 more = 80, and then the
remaining ones to get to 82; 80 + 2 = 82. The total difference is
4+10+10+10+10+10+2 = 56. Students will eventually begin to add all of the tens
in one chunk, 30 + 50 = 80
OR
Students may add tens to get close and then add ones. 26 + 10 +10 +10 +10 =
76, 76 + 4 = 80, 80 + 2 = 82. This evolves into adding a group of tens and then
ones, 26 + 50 = 76, 76 + 4 = 80, 80 + 2 = 82; 56 is add to 26 to make 82.
OR
Students may overshoot by adding tens and then remove the extra ones. 26 + 60
= 86, 86 – 4 = 82, 60 – 4 = 56.
Possible Models
Number Line
Number Line
How many from 26 to 82?
How many from 26 to 82?
+4 +10 +10 +10 +10 +10 +2 = 56

26 30 40 50 60
70
80 82
5
15
25
35
45
55
65
75
85
6
16
26
36
46
56
66
76
86
+4
+50
2
12
22
32
42
52
62
72
82
3
13
23
33
43
53
63
73
83
4
14
24
34
44
54
64
74
84
7
17
27
39
47
57
67
77
87

How many more to make 82?


56 more makes 82
+2 = 56

26 30
80 82
Hundreds Chart
1
11
21
31
41
51
61
71
81
Base Ten Blocks
Start with 26
8
18
28
38
48
58
68
78
88
9
19
29
39
49
59
69
79
89
10
20
30
40
50
60
70
80
90
How many from
26 to 82?
Down 5 to 76
(+50), then right 6
to 82 which
requires wrapping
to the next line
(+6)
56
Possible Recording
Schemes 26 + 4 + 50 + 2 = 82
26
+4
+10
+10
+10
+10
+10
+ 3
82
56
EXAMPLES OF SUBTRACTION STRATEGIES
COUNTING BACK
Students who have primarily been exposed to take away subtraction problems will be
drawn to this strategy. Students may begin at the whole and find the difference
between the whole and the part being subtracted or begin at the whole and count back
the number being subtracted.
Example: 82 - 26
Students may begin at the whole and count back tens, 82 -10 - 10 = 62, and count
back the remaining ones. When counting back the ones students may count back
one at a time or in chunks, 62 – 2 = 60; 60 – 4 = 56.
OR
Students may count back to the next benchmark, then remove the tens, then
remove the remaining ones, 82 – 2 = 80; 80 – 20 = 60; 60 – 4 = 56
OR
Students may overshoot when taking off tens and then compensate by adding the
amount they overshot, 82 – 30 = 52; 4 too many were subtracted so add 4,
52 + 4 = 56. Number Line
Number Line
Number Line
Count back tens, then ones
-4 -2
-40
  56

60 62
82
Count back ones, then
tens, then ones
-4
-40
-2


56 60
80 82
Overshoot and then
compensate
+4
-30


52 56
82
KEEPING A CONSTANT DIFFERENCE (ALSO REFERED TO AS COMPENSATION)
This strategy involves adjusting both numbers in the problem by the same amount in
order to make a new problem that is easier to solve.
Example: 34-19
A student may recognize that subtracting 20 is easier than subtracting 19 and add
one to both numbers, making the problem 35 – 20 = 15.
MAKE EASY NUMBERS (ALSO REFERED TO AS COMPENSATION)
This strategy is similar to KEEPING A CONSTANT DIFFERENCE, but only one of the
numbers is altered and then the difference is altered to compensate. Frequently, but
not always, the number being subtracted is altered to create a problem in which a
multiple of ten is being subtracted.
Example: 62 – 37
A student may find it easier to subtract 40 instead of subtracting 37. The extra 3
that was subtracted would then need to be added to the difference. 62 – 40 = 22,
then add the 3, 22 + 3 = 25.
Example: 86 – 23
A student may prefer subtracting 20 over subtracting 23 and alter this problem by
subtracting 3 from 23. The new problem would be 86 – 20 = 66. In this case 3 too
few were subtracted and must be subtracted from the difference. 66 - 3 = 63
EXPANDED FORM
Example: 849-145 (Decompose only the subtrahend)
849 - (100 + 40 + 5) =
849 - 100 = 749
749 - 40 = 709
709 - 5 = 704