Addition and Subtraction in the Early Grades (K-2)

Overview of Grades K–2

Students develop meanings for addition and subtraction as they encounter problem situations in Kindergarten, and they extend these meanings as they encounter increasingly difficult problem situations in Grade 1. They represent these problems in increasingly sophisticated ways. And they learn and use increasingly sophisticated computation methods to find answers. In Kindergarten, teachers help children lay the foundation for understanding the base-ten system by drawing special attention to 10. Children learn to view the whole numbers 11 through 19 as ten ones and some more ones. In first grade, students learn to view ten ones as a unit called a ten. The ability to compose and decompose this unit flexibly and to view the numbers 11 to 19 as composed of one ten and some ones allows development of efficient, general base-ten methods for addition and subtraction. Students see a two-digit numeral as representing some tens and they add and subtract using this understanding. At Grade 2, students extend their base-ten understanding to hundreds. They now add and subtract within 1000, with composing and decomposing, and they understand and explain the reasoning of the processes they use. They become fluent with addition and subtraction within 100. In each grade, the situations, representations, and methods are calibrated to be coherent and to foster growth from one grade to the next.

Developed using the following resources: Danielle Seabold, Kalamazoo RESAhttp://www.ncpublicschools.org/acre/standards/common-core-tools/ Matthew Mayer, Kalamazoo RESAhttp://ime.math.arizona.edu/progressions/ Angela Justice, Kalamazoo Public Schoolshttp://www.turnonccmath.net/

Kindergarten – Adding and Subtracting within 10Standard Explanation Build-It Draw-It Write-It

K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

Students act out adding and subtracting by representing quantities in the situation. To do this, students must mathematize a real-world situation, focusing on the quantities and their relationships rather than non-mathematical aspects of the situation. Situations can be acted out and/or presented with pictures or words. Math drawings facilitate reflection and discussion because they remain after the problem is solved.

Represent addition and subtraction with objects, fingers

Represent addition and subtraction with mental images, drawings, sounds (e.g., claps), acting out situations

Represent addition and subtraction with verbal explanations, expressions, or equations.

Common Core State Standards for Mathematics states, “Kindergarten students should see addition and subtraction equations, and student writing of equations in kindergarten is encouraged, but it is not required.” Please note that it is not until First Grade when “Understand the meaning of the equal sign” is an expectation (1.OA.7). Therefore, before introducing symbols (+, -, =) and equations, kindergarteners require numerous experiences using joining (addition) and separating (subtraction) vocabulary in order to attach meaning to the various symbols. For example, when explaining a solution, kindergartens may state, “Three and two is the same amount as 5.”

K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Put Together/Take Apart situations with Both Addends Unknown play an important role in Kindergarten because they allow students to explore various compositions that make each number. As students decompose a given number to find all of the partners that compose the number, the teacher can record each decomposition with an equation such as 5 = 4 + 1, showing the total on the left and the two addends on the right. Students can find patterns in all of the decompositions of a given number and eventually summarize these patterns for several numbers. If equations are used, a mathematical representation (picture, objects) needs to be present as well.

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects.

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using drawings, and record each decomposition by a drawing.

Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a equation.

Q: “Bobby Bear is missing 5 buttons on his jacket. How many ways can you use blue and red buttons to finish his jacket? Draw a picture of all your ideas. Students could draw pictures of:4 blue and 1 red button 3 blue and 2 red buttons2 blue and 3 red buttons1 blue and 4 red buttons

Kindergarten – Adding and Subtracting within 10Standard Explanation Build-It Draw-It Write-It

K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

In Standard K.OA.3 earlier, students became proficient with decomposing numbers less than or equal to 10 into different combinations. In this standard students expand on this proficiency and, when given any number from 1 to 9, find the missing part that when added (joined) to that number makes 10. Students use a variety of strategies including objects or drawings to help them solve this type of problem and record their answers with a drawing or an equation.

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects.

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using drawings, and record the answer with a drawing.

For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., record the answer with an equation.

“A full case of juice boxes has 10 boxes. There are only 6 boxes in this case. How many juice boxes are missing?

K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

As students become proficient at using strategies and modeling for addition and subtraction, they use these to solve word problems involving adding and subtracting within 10. One way to present these types of problems is using a variety of types of “join/separate” problems.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using drawings to represent the problem.

Does Not Apply

Six crayons are in the box. Two are red and the rest are blue. How many blue crayons are in the box?

 Length-based models (e.g., lengths of connecting cubes and number lines).

K.OA.5 Fluently add and subtract within 5.

Later in the year, students solve addition and subtraction equations for numbers within 5, for example 2 + 1 = □ or 3 – 1 = □. While still connecting these equations to situations verbally or with drawings. .

Fluently add and subtract within 5 using representations.

Fluently add and subtract within 5.

Using 5 frames or other objects can be used to help develop fluency.

Students are fluent when they display accuracy, efficiency (a reasonable amount of steps in about 3-5 seconds without resorting to counting), and flexibility (using strategies such as the distributive property).

Kindergarten – Adding and Subtracting within 10Standard Explanation Build-It Draw-It Write-It

K.NBT.1: Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

Finally, composing and decomposing numbers from 11-19 into ten ones and some further ones builds from all of this work. This is a vital first step kindergarteners must take toward understanding base-ten notation for numbers greater than 9.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, by using objects.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, by using drawings, and record each composition or decomposition by a drawing.

Compose and decompose numbers from 11 to 19 into ten ones and some further ones, record each composition or decomposition by an equation.

First Grade – Adding and Subtracting within 100Standard Explanation Build-It Draw-It Write-It

1.OA.4 Understand subtraction as an unknown-addend problem.

The meaning of subtraction as an unknown-addend addition problem is one of the essential understandings students will need in middle school in order to extend arithmetic to negative rational numbers.

Understand subtraction as an unknown-addend problem by using objects.

Understand subtraction as an unknown-addend problem by using drawings.

Understand subtraction as an unknown-addend problem.

1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

Grade 1 students are extending the range of problem types and subtypes they can solve, they are also extending the range of numbers they deal with. Students know, with fluency and automaticity, all the addition and subtraction number combinations from 1 to 10. With the aid of models and other strategies, they add and subtract numbers within 20

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies with objects.

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies with drawing.

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

First Grade – Adding and Subtracting within 100Standard Explanation Build-It Draw-It Write-It

1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

As students get used to extending the range of problem types and subtypes they can solve, they are also extending the range of numbers they deal with and the sophistication of the methods they use to add and subtract within this larger range.

Use addition and subtraction within 20 to solve word problems by using objects.

Use addition and subtraction within 20 to solve word problems by using drawings.

Use addition and subtraction within 20 to solve word problems by using equations with a symbol for the unknown number to represent the problem.

First Grade – Adding and Subtracting within 100Standard Explanation Build-It Draw-It Write-It

1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

As students get used to extending the range of problem types and subtypes they can solve, they are also extending the range of numbers they deal with and the sophistication of the methods they use to add and subtract within this larger range.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers by using objects.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers by using drawings.

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers.

1.OA.3 Apply properties of operations as strategies to add and subtract.

Students solve problems using the commutative, associative, and identity properties and apply them as strategies to solve problems.The commutative property of addition states that a + b = b + a, for any numbers a and b.The associative property of addition states that (a + b) + c = a + (b + c)

Apply properties of operations as strategies to add and subtract using objects.

Apply properties of operations as strategies to add and subtract using drawings.

Apply properties of operations as strategies to add and subtract.

Students also learn to apply more than one of the properties at the same time. For example, to solve 2 + (5 + 8), students apply both the commutative and associative properties. They first use the commutative property to reason that 2 + (5 + 8) = 2 + (8 + 5) and the associative property to reason the following: 

2 + (8 + 5) = (2 + 8) + 5            = 10 + 5

     = 15

 

First Grade – Adding and Subtracting within 100Standard Explanation Build-It Draw-It Write-It

1.OA.2 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

A known addition or subtraction problem can be used to solve a related addition or subtraction problem by decomposing one addend and composing it with the other addend. 

Students extend their abilities to solve addition word problems (e.g., “joining and separating problems”) with three numbers (whose sum is less than or equal to 20) and with the unknown in any position.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using drawings.

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using equations with a symbol for the unknown number to represent the problem.

Mrs. Smith has 4 oatmeal raisin cookies, 5 chocolate chip cookies, and 6 gingerbread cookies. How many cookies does Mrs. Smith have?

1.NBT.4 Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 

Using models, drawings, and their knowledge of place value, students add and subtract within 100.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models and to develop strategies.

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using drawings to develop strategies. 

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

2.OA.1 Use addition and subtraction within 100 to solve one- andtwo-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

Students represent and solve situational problems of all three types which involve addition and subtraction within 100 rather than within 20, and they represent and solve two-step situational problems of all three types.

Use addition and subtraction within 100, by using objects.

Use addition and subtraction within 100, by using drawings.

Use addition and subtraction within 100, by using equations with a symbol for the unknown number to represent the problem.

“There are 54 fans at a little league baseball game. After the 2nd inning, 12 fans left the game. After the 3rd inning, 28 new fans arrived. How many fans were at the baseball game after the 3rd inning?”

2.OA.2 Fluently add and By the end of the K–2 grade Fluently add and subtract Fluently add and subtract Fluently add and subtract within 20 using

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

span, students have sufficient experience with additionand subtraction to know single-digit sums from memory; as should be clear from the foregoing, this is not a matter of instilling facts divorced from their meanings, but rather as an outcome of a multi-year process that heavily involves the interplay of practice and reasoning.

within 20 using mental strategies. Use objects to help with modeling.

within 20 using mental strategies. Use drawings to help with strategy development.

mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g.,by using drawings (such as drawings of rulers) and equations witha symbol for the unknown number to represent the problem.

In Grades 2 and 3, students continue and extend their work with adding and subtracting situations to length situations.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawing.

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using equations with a symbol for the unknown number to represent the problem.

2.MD.6 Represent whole numbers as lengths from 0 on

In Grades 2 and 3, students continue and extend their work

Represent whole numbers as lengths from 0 on a

Represent whole numbers as lengths from 0 on a number

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, . . ., and represent whole-number sums and differences within 100 on a number line diagram.

with adding and subtracting situations to length situations.

number line diagram by building a number line then placing number cards on the number line.

line diagram by writing in numbers and spacing on the number line.

2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Rectangular arrays provide opportunities for students to apply strategies of counting, skip counting, and addition and subtraction as they transition toward multiplication.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns. Identify the addends.

Use addition to find the total number of objects, drawn or pictures, arranged in rectangular arrays with up to 5 rows and up to 5 columns. Identify the addends.

Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Using place value strategies (by breaking or making hundreds, tens and ones), the properties of operations that they learned in Standard 1.OA.3 (e.g.,the commutative, associative, and identity properties of addition), and the inverse relationship between addition and subtraction, students fluently add and subtract sums up to 100.  

Fluently add and subtract within 100 using objects

Fluently add and subtract within 100 using drawings.

Fluently add and subtract within 100 using strategies to understand equations.

Again, to promote fluency beyond rote memorization, students are encouraged to explain and justify their answers verbally, using writing, and equations.

2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

Students explain why addition and subtraction strategies work, both verbally and written, by relating their strategies to place value and the properties of operations (e.g., the commutative, associative, and identity properties of addition).

Explain why addition and subtraction strategies work, using objects.

Explain why addition and subtraction strategies work, using drawings.

Explain why addition and subtraction strategies work, using place value and the properties of operations.

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.

Students use properties of addition (e.g., the commutative, associative, and identity properties of addition) to add up to four two-digit whole numbers. They create and use increasingly sophisticated strategies based on these properties (e.g., “making tens”) to solve addition and subtraction problems. By comparing a variety of solution strategies, students relate addition and subtraction as inverse operations that are connected to one another.

Add up to four two-digit numbers using objects.

Add up to four two-digit numbers using drawings.

Add up to four two-digit numbers using strategies based on place value and properties of operations.

Second Grade – Adding and Subtracting within 1,000Standard Explanation Build-It Draw-It Write-It

2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Students continue to use the strategies (place value, composition and decomposition), models, and properties (commutative, associative, and identity properties of addition) that they learned in earlier grades (see Standards 1.NBT.4 and 1.OA.6) to solve more complex addition and subtraction problems (three-digit numbers, and sums up to 1000.) Students explain their place value strategies as well as composing and decomposing tens or hundreds.

Add and subtract within 1000, using concrete models.

Add and subtract within 1000, using drawings.

Add and subtract within 1000, relate the strategy to a written method.

Summary

Kindergarten: Students in Kindergarten work with three kinds of problem situations: Add To with Result Unknown; Take From with Result Unknown; and Put Together/Take Apart with Total Unknown and Both Addends Unknown. The numbers in these problems involve addition and subtraction within 10. Students represent these problems with concrete objects and drawings, and they find the answers by counting (Level 1 method). More specifically,

For Add To with Result Unknown, they make or draw the starting set of objects and the change set of objects, and then they count the total set of objects to give the answer.

For Take From with Result Unknown, they make or draw the starting set and “take away” the change set; then they count the remaining objects to give the answer.

For Put Together/Take Apart with Total Unknown, they make or draw the two addend sets, and then they count the total number of objects to give the answer.

Grade 1: Students in Grade 1 work with all of the problem situations, including all subtypes and language variants. The numbers in these problems involve additions involving single-digit addends, and the related subtractions. Students represent these problems with math drawings and with equations. Students master the majority of the problem types. They might sometimes use trial and error to find the answer, or they might just know the answer based on previous experience with the given numbers. But as a general method they learn how to find answers to these problems by counting on (a Level 2 method), and they understand and use this method. Students also work with Level 3 methods that change a problem to an easier equivalent problem. The most important of these Level 3 methods involve making a ten, because these methods connect with the place value concepts students are learning in this grade (see the NBT Progression) and work for any numbers. Students also solve the easier problem subtypes with these Level 3 methods.

The four problem subtypes that Grade 1 students should work with, but need not master, are: Add To with Start Unknown Take From with Start Unknown Compare with Bigger Unknown using “fewer” language (misleading language suggesting the wrong operation) Compare with Smaller Unknown using “more” language (misleading language suggesting the wrong operation)

Grade 2: Students in Grade 2 master all of the problem situations and all of their subtypes and language variants. The numbers in these problems involve addition and subtraction within 100. They represent these problems with diagrams and/or equations. For problems involving addition and subtraction within 20, more students master Level 3 methods; increasingly for addition problems, students might just know the answer (by end of Grade 2, students know all sums of two-digit numbers from memory). For other problems involving numbers to 100, Grade 2 students use their developing place value skills and understandings to find the answer. Students work with two-step problems, especially with single-digit addends, but do not work with two-step problems in which both steps involve the most difficult problem subtypes and variants.

Common Addition and Subtraction Situations