Addition and Subtraction
Computational Fluency Resources
Websites
http://www.k-5mathteachingresources.com/addition-and-subtraction-activities.html
 Basic Levels for addition and subtraction with activities to support students at all levels of addition
and subtraction.
https://www.heinemann.com/shared/onlineresources/E02963/oconelladd.pdf
 An excerpt from a book that talks about the different subtraction and addition skills. It also gives
many examples of each type of strategy.
http://manatee.sp.brevardschools.org/Smith/Class%20Documents/Strategy%20Posters.pdf
http://bridges1.mathlearningcenter.org/resources/blog/2009/10/addition-and-subtraction-fact-strategy-posters
 Two sets of classroom posters that simply explain different addition and subtraction strategies.
https://utahelementarymath.wordpress.com/2014/03/27/strategies-for-addition-and-subtraction/
 Article explaining different addition and subtraction strategies and the reasoning behind them.
http://www.cpalms.org/Public/PreviewResource/Preview/36525
 Sample formative assessments with guidance on next steps based on each assessment.
http://www.showme.com/sh/?h=BhO20g4
 Video showing an example of using an open number line to add
http://www.k-5mathteachingresources.com/empty-number-line.html
 “What is an Empty [Open] Number Line?” explains how to introduce and support students as they
use open number lines to add and subtract increasingly larger numbers.
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
Foundation Facts
+1/+2
Students build on their understanding of counting by exploring 1 or 2 more and 1
or 2 less.
+0
Using their knowledge of the concept of addition, students explore what happens
when they add or subtract nothing from a quantity.
+10
Adding 10 to a single-digit number results in a 2-digit sum. Students explore
adding 10 in order to build understanding and automaticity that will be needed
later when exploring the using-ten strategy.
Doubles
Students explore the concept of doubling and what it means to add 2 groups of
equal size.
Building on the Foundation
Making Ten
Because 10 is foundational in our number system, students explore the different
ways in which 2 addends result in a sum of 10. This knowledge becomes critical as
they later explore using tens to find unknown facts. Building on the Foundation
Using tens Now that students know combinations of addends that have a sum of
10, they use their understanding of the flexibility of numbers to find ways to break
apart addends to create simpler facts by using tens (e.g., 9 + 7 is changed to 10 +
6).
Using
Doubles
Students’ knowledge of doubles facts is now put to use to find unknown facts that
are near-doubles (e.g., 4 + 5 might be thought of as 4 + 4 + 1). Figure 1. This
suggested teaching sequence begins with simpler facts and then connects each
new set of facts to students’ previous experiences.
Source: https://www.heinemann.com/shared/onlineresources/E02963/oconelladd.pdf
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
3 Levels of Math Understanding
Level One: Direct Modeling by Counting All or Taking Away
Level Two: Counting On
Level Three: Convert to an Easier Equivalent Problem
Reference: Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics, K,
Counting and Cardinality; K-5 Operations and Algebraic Thinking. Tucson, AZ: Institute for Mathematics and Education, University of
Arizona.
Examples of Level Two Strategies

Counting on: 8 + 4 = □ (8 …9, 10, 11, 12)

Counting back: 12 – 4 = □ (12…11, 10, 9, 8)
Examples of Level Three Strategies

Making tens: 5 + 7 = □ (5 = 2 + 3 so 3 + 7 = 10, then 10 + 2 = 12)

Doubles: 6 + 6 = □

Doubles plus/minus one: 6 + 7 = □• (6 + 6 + 1 or 7 + 7 – 1)

Decomposing a number leading to a ten: 15 – 7 = □, so 15 – 5 = 10, then 10 – 2 = 8)

Working knowledge of fact families/related facts/number bonds: 3 + 9 = 12 so 12 – 9
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
Opportunistic Strategies for Addition (Level 3 Thinking)
These strategies build on students’ knowledge of place value, number sense, and properties of operations to
perform calculations more flexibly and efficiently, sometimes without pencil and paper. These are important
strategies to explore with students, in addition to their work leading to the standard algorithm for addition.
Place Value
Start adding at the left in a vertical problem, rather than at the right, as follows:
Students add 40 + 30 to get 70. Then they add 7 + 3 to get 10. Then they find the sum 70 + 10 = 80.
Breaking Apart (Place Value), also known as “Decomposing” or “Splitting”
a. Break both numbers down to place value and add each, starting with the largest.
46 + 25 = 
46 breaks into 40 plus 6 (40 + 6), 25 breaks into 20 plus 5 (20 + 5)
40 + 20 = 60
6 + 5 = 11
60 + 11 = 71
OR
b. Keep one number intact and only break second number down by place value and adding each place:
46 + 25 = 
46 stays intact and 25 breaks into 20 and 5
46 + 20 = 66
66 + 5 = 71 or *66 + (4 + 1)
*Note: some students may prefer to break the 5 apart (4 + 1) so that they can add 4 to 66 and get 70, then add on 1.
Compensation
Round one or more of the numbers to numbers that are easier to work with, then compensate:
256 + 687
-13
256 + 700 = 956
round 687 to 700, will have to take away 13
956 – 13 =
subtract the 13, by decomposing it to 10 + 3
956 – 10 = 946
946 – 3 = 943
*****13 is added to 687 to get 700, an easier number to work with - keeping track of the adjustment is critical
to making this strategy work—encourage students to box the adjustment (here we box the adjustment as -13
since 13 was added, now 13 must be subtracted out)
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
Opportunistic Strategies for Addition
These strategies build on students’ knowledge of place value, number sense, and properties of operations to
perform calculations more flexibly and efficiently, sometimes without pencil and paper. These are important
strategies to explore with students, in addition to their work leading to the standard algorithm for addition.
Transformation
Transform the problem into an equivalent problem that is easier (like compensation, this is a strategy more
advanced math thinkers can handle, you’re adding to one and taking away the same amount from the other)
46 + 28 = ___
28 + 2 = 30
46 – 2 = 44
30 + 44 = 74
adding 2 to 28 makes it 30, an easy number to work
if 2 is added into this equation, then 2 must be subtracted from the 46
256 + 687 = 
256 + 700 = 956
add 13 to 687 to make it 700
256 – 13 = 243
subtract 13 from 256 to make it 243
700 + 243 = 943
Open Number Line
The number line is used as a tool to help build students’ understanding of the base-ten number system, and to
solve problems using addition and subtraction within 100 and within 1000. You will want to move students to
use this tool to count by 10s when large numbers and/or large gaps in an equation are present. One of the
interesting things about mental calculations is that we do not all think the same way. The empty number line
allows students to see the variety of ways that the same question can be solved.
A sunflower is 47 cm tall. It grows another 25cm. How tall is it?
Source: http://www.k-5mathteachingresources.com/empty-number-line.html
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
Opportunistic Strategies for Subtraction (Level 3 Thinking)
These strategies build on students’ knowledge of place value, number sense, and properties of operations to
perform calculations more flexibly and efficiently, sometimes without pencil and paper. These are important
strategies to explore with students, in addition to their work leading to the standard algorithm for subtraction.
Decomposing/Breaking Apart
Subtract one number—in parts—from the other number, which stays intact, always starting with largest place
value to subtract.
a. 54 – 23 = 
23 can be broken into 20 + 3
54 – 20 = 34
34 – 3 = 31
b. 56 – 29 = 
29 can be broken into 20 + 6 + 3, breaking 9 into 6 + 3 makes it easier to subtract
56 – 20 = 36
36 – 6 = 30
30 – 3 = 27
c. 547 – 297 = 
keep 547 intact, break 297 into 200 + 90 + 7, subtract out one place value at a time
547 – 200 = 347
347 – 90 = 257
257 – 7 = 250
OR
547 – 297 =
break 297 into 247 + 50, subtract out each part
547 – 247 = 300
300 – 50 = 250
Adding up/Counting On
Start with smaller number, add up to a landmark number*, from the landmark add up to get to target number.
a. 212 – 197 = 
197 + 3 = 200*
200 + 12 = 212
3 + 12 = 15
Add the two numbers you used.
b. 516 – 305 = 
305 + 195 = 500*
500 + 16 = 516
195 + 16 = 211
Add the two numbers you used.
Adding up is a good strategy when one of the subtrahends involves 0s. Students have a great deal of difficulty
subtracting across zeros with the traditional ungrouping/borrowing algorithm.
c. $10.00 – $4.75 = 
$4.75 + $.25 = $5.00*
$5.00 + $5.00 = $10.00
$5.00 + $.25 = $5.25
Created 7/2015
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Addition and Subtraction
Computational Fluency Resources
Opportunistic Strategies for Subtraction (Level 3 Thinking)
These strategies build on students’ knowledge of place value, number sense, and properties of operations to
perform calculations more flexibly and efficiently, sometimes without pencil and paper. These are important
strategies to explore with students, in addition to their work leading to the standard algorithm for subtraction.
Subtracting from 9s
Subtracting from 9s doesn’t require ungrouping/regrouping/borrowing, which can be problematic for students.
a. 1,000 – 273 = 
1,000 = 999 + 1 box the adjustment to remember to add it back in
999
-273
----------726 + 1 = 727
now add back the 1, the answer is 727
b. 1006 – 273 = 
1,006 = 999 + 7 box the adjustment to remember to add it back in
999
– 273
__________
726 + 7 = 733 now add back the 7, making the answer 733
Compensation
Adjust one of the numbers in a math problem in order to make the numbers easier to work with.
a. 45 – 27 = 
27 – 2 = 25
subtract 2 to get to 25, an easy number to subtract from 45—box the adjustment for later
45 – 25 = 20
20 – 2 = 18
you need to subtract 2 more to subtract a total of 27
b. 834 – 237 = 
237 – 3 = 234
subtract 3 to get to 234, a friendly number to subtract from 834—box the adjustment for later
834 – 234 = 600
600 – 3 = 597
you need to subtract 3 more to subtract a total of 237
b. 361 – 55 = 
55 + 6 = 61
361 – 61 = 300
300 + 6 = 306
add 6 to get to 61, an easy number to subtract from 361—box the adjustment for later
you need to add 6 more because you took away 6 too many
Note: This strategy can become confusing for students, when determining whether to subtract out or add in the
compensated numbers. Use of visual models or connections with other strategies may help to develop understanding.
Created 7/2015
Page 7 of 8
Addition and Subtraction
Computational Fluency Resources
Opportunistic Strategies for Subtraction (Level 3 Thinking)
These strategies build on students’ knowledge of place value, number sense, and properties of operations to
perform calculations more flexibly and efficiently, sometimes without pencil and paper. These are important
strategies to explore with students, in addition to their work leading to the standard algorithm for subtraction.
Open Number Line
The number line is used as a tool to help build students’ understanding of the base-ten number system, and to
solve problems using addition and subtraction within 100 and within 1000. You will want to move students to
use this tool to count by 10s when large numbers and/or large gaps in an equation are present. One of the
interesting things about mental calculations is that we do not all think the same way. The empty number line
allows students to see the variety of ways that the same question can be solved.
a. I need 72 dollars to buy a skateboard. I have 39 dollars already. How many more dollars do I need to save?
39 is placed near the start of the empty number line and 72 near the end. We can count up in 'friendly'
jumps to reach 72. First a jump of 1 to reach 40 (multiples of ten are easy numbers to jump to and from),
then a jump of 30 to reach 70 and finally a jump of 2 to reach our target of 72. I need to save 33 more
dollars.
b. There are 543 people on the subway platform. 387 board a train. How many people are left on the platform?
We start at 387 and count up in 'friendly' jumps to reach 543. First a jump of 13 to reach 400 (multiples of
ten and hundred are easy numbers to jump to and from) then a jump of 100 to reach 500 and finally a
jump of 43 to reach the target of 543. 156 people are left on the platform.
Source: http://www.k-5mathteachingresources.com/empty-number-line.html
Some strategies taken from: Parrish, Sherry. 2010. Number Talks: Helping Children Build Mental Math and Computation
Strategies. Sausalito, CA: Math Solutions.
Created 7/2015
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