Single Digit Addition and Subtraction
Teacher Goal:
Compute basic addition facts and the related subtraction facts using strategies.
After receiving a University Degree in Education, one would assume that they would be prepared to teach simple addition and subtraction to 6 year olds. However, frustrational levels may arise when students are unable to grasp such a “simple” concept. Teachers must learn how students think and be able to use student responses to guide their instruction.
Teacher Strategies
There are many strategies that can be used to help students discover the relationships between numbers. Students are capable of inventing their own strategies to solve different problem types. There are different levels students travel through as they develop understanding of the different concepts. The first level is the use of direct modeling using manipulatives and pictures of physical objects. Direct modeling is the use of these physical objects to show the action and relationships of the numbers in various problems. As understanding develops, students eliminate the need for direct modeling and invent more efficient strategies. These strategies may include counting on and counting down using number lines or finger counting. Once students have a more complete understanding of the concepts, they will replace these strategies with number facts (Bahr, 2010).
Direct Modeling –> Counting –> Number Facts
Direct Modeling Strategies 
Counting Strategies 
Joining All Joining To Trial And Error Separating From Matching 
Counting On Counting On To Counting Down Separating From Matching

Direct Modeling Manipulatives
Counting Strategies
Instruction
When planning instruction, the teacher needs to define the number range each student works in. This can be found by presenting the students with different types of problems and number ranges. For example, when given the problem 5+2=7, the student may use number facts, but when given a problem such as 8+9=17, the student may revert back to direct modeling or counting strategies. Teachers can then use this information in developing the types of problems to guide instruction. The different types of problems are shown below. Number ranges can be adjusted to the students needs.
Join 

Result Unknown 
Change Unknown 
Start Unknown 

Joy has (3, 4, 7) flowers for her bouquet. Alice gave her (5, 8, 9) more flowers. How many flowers does she have altogether? 3 + 5 = _(semantic) 4 + 8 = _ 7 + 9 = _ 3 + 5 = _(computational) 4 + 8 = _ 7 + 9 = _ 
George has (3, 4, 7) flowers. How many more does he need to have (8, 12, 16) flowers? 3 + _= 8 (semantic) 4 + _= 12 7 + _= 16 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 
Abby had some flowers. John gave her (3, 4, 7) more flowers. Now she has (8, 12, 16). How many did Abby start with? _+ 3 = 8 (semantic) _+ 4 = 12 _+ 7 = 16 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 

Separate 

Result Unknown 
Change Unknown 
Start Unknown 

Ron had (8, 12, 16) flowers. He gave (3, 4, 7) to Kara. How many does he have left? 8 – 3 = _(semantic) 12 – 4 = _ 16 – 7 = _ 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 
Rachel had (8, 12, 16) flowers. She gave some to Carson. Now she has (3, 4, 7) left. How many did she give to Carson? 8 – _= 5 (semantic) 12 – _= 8 16 – _= 7 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 
Alice has some flowers. She gave (3, 4, 7) to George. Now she has (5, 8, 9) left. How many flowers did Alice start out with? _– 3 = 5 (semantic) _– 4 = 8 _– 7 = 9 3 + 5 = _(computational) 4 + 8 = _ 7 + 9 = _ 

PartPartWhole 

Whole Unknown 
Part Unknown 

Ron has (3, 4, 7) white flowers and (5, 8, 9) yellow flowers. How many flowers does he have? 3 + 5 = _(semantic) 4 + 8 = _ 7 + 9 = _ 3 + 5 = _(computational) 4 + 8 = _ 7 + 9 = _ 
Joy had (8, 12, 16) flowers. (3, 4, 7) purple and the rest were pink. How many pink flowers does Joy have? 8 = 3 + _ (semantic) 12 = 4 + _ 16 = 7 + _ 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 

Compare 

Difference Unknown 
Compare Quantity Unknown 
Referent Unknown 

George has (3, 4, 7) flowers and John has (8, 12, 16) flowers. How many more flowers does John have thank George? 3 + _= 8 (semantic) 4 + _= 12 7 + _= 16 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _ 
Carla has (3, 4, 7) flowers. Sean has (5, 8, 9) more flowers than Carla. How many flowers does Sean have? 3 + 5 = _(semantic) 4 + 8 = _ 7 + 9 = _ 3 + 5 = _(computational) 4 + 8 = _ 7 + 9 = _ 
Megan has (8, 12, 16) flowers. She has (3, 4, 7) more than Tanner. How many flowers does Tanner have? 8 = 3 + _ (semantic) 12 = 4 + _ 16 = 7 + _ 8 – 3 = _(computational) 12 – 4 = _ 16 – 7 = _

Back to the Basics
Generalizations
Students may independently create generalizations to facilitate computation. These generalizations support recall of number facts and often assist students in flexible thinking strategies. We want to make certain the student’s generalizations are accurate, although specifically tell them rules to follow.
Developmental steps of generalization:
 Kinesthetic experiences
 Visual representation with the use of a model
 Symbolic representation as a visual
 Creating a model in the head, mental calculations
 Computing mentally with use of generalization
We should encourage effective generalizations:
 Combinations of number to ten
 Communtativity
 One more or one less
 Doubles or near doubles
 Making tens
 10 and some more
 10 more or less
Commutativity
The commutative property of addition is the understanding that numbers can be added in any order without changing the answer. The purpose of identifying this property is to show students that they can start addition problems with the number they feel most comfortable with. Students use this property without knowing its definition as they subconsciously choose which number to begin with. This property becomes important as their number ranges increase. For example, in the problem 42+7=49, while they may not be comfortable starting with 42, it will be easier for them to start with the larger number.
Doubles and NearDoubles
Most students are able to learn and recall doubles easily compared to other math facts. Therefore using doubles as a strategy allows students to easily solve problems that are near doubles by adding or subtracting 1. If 3+3=6, and 3+3+1=7, then 3+4=7.
Making tens
There are ten steps students go through when generalizing how to make tens. They are:
 Recognize individual quantities of the 10 frames
 Knows 10 and some more
 Counts on for 9 and some more
 Moves counter to make ten
 Moves dot mentally to make ten
 Moves mentally when only sees 9 and hears another number
 Moves dot mentally when hearing both numbers
 Generalizes a rule for adding 9
 May revert to counting on when introducing 8 and some more and 7 and some more
 Generalizes the idea of breaking apart numbers to add two digit numbers
(Bahr, 2010)
Resources for Kids
http://mathforum.org/templepow/support/koline.html
http://www.learningbox.com/Base10/CatchTen.html
http://www.mrsjonesroom.com/themes/math.html
http://funschool.kaboose.com/formulafusion/games/game_dr_brains_robot.html
http://www.uen.org/core/linkList.do?courseNum=5010&itemId=2938
Resources for Teachers and Parents
http://www.uen.org/core/lessonList.do?courseNum=5010&itemId=2938
http://www.teachnology.com/themes/math/addsub/
http://www.ldonline.org/spearswerling/9655
http://www.homeschoolmath.net/online/addition_subtraction.php
http://www.theteacherscafe.com/Worksheets/Math/AdditionSubtraction/AdditionSubtraction.htm