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Finding Addition Patterns

Number and Operations
Grace M. Burton
Location: unknown

A game encourages students to find the sums of two one-digit numbers. Students explore commutativity and examine patterns on an addition table. They then use a personal addition chart to record and keep track of known facts.

Begin the lesson by modeling how to complete number patterns which involve addition. For example, give the students the pattern "2, 5, and 8," and ask them to name the next number in the pattern [11.]. Ask students what the rule is. [Add 3 to the previous number.] Repeat this with other addition number patterns, such as 3, 6, 9, ____ or 4, 8, 12, ____.

Then give each student crayons and a copy of the Hundreds Chart Activity Sheet. Ask the students to find as many numbers as they can whose digits add up to 3 and color the numbers blue: 3, 12, 21, 30.

pdficonHundreds Chart Activity Sheet 

When they are ready, ask them to look for and color any patterns involving addition of the digits in the numbers. Allow them a few minutes to do this, and then invite them to share the patterns they found. An example that shows one possible pattern for adding the digits is in pink in the chart below.

Color Pattern_Finding Addition Patterns 

As a challenge activity, invite the students to discover the pattern shown in yellow in the chart above. [The sum of the digits increases by 1.] You might have students gather around computers to observe this chart, project the chart using a projection device, make a color transparency of the chart and project it on an overhead, or provide individual color copies for groups of students.

Ask a volunteer to show the sum of 5 + 6, record it, then enter 6 + 5, record that sum, and compare the results. Write the two equations on the board. Encourage the students to try reversing the addends with other pairs of numbers. Have the students describe their observations orally. Finally, introduce the students, if needed, to the formal term, "commutative," and discuss the meaning of this concept. Ask them how knowing this concept can help in learning number facts. For example, if they know that 3 + 4 = 7, then they also know that 4 + 3 = 7.

Assign the students to groups of three to five each, and give each group two number cubes. Have the students each take one turn rolling the number cubes and naming the sum. Encourage them to name the sum from memory if they can, or they may use counters or a calculator if it is a sum they do not know.

Have each student write his or her sum on a piece of paper. When all the students have recorded their sum, ask the member of each pair with the highest sum to draw a circle around it. If more than one student has that number, each should circle it.

Allow the students to play several rounds. Then have them count the number of sums they circled. The one with the most circled sums is the winner of the game. Ask the first four Questions for Students, in the Questions + Reflection tab.

To conclude the lesson, display an addition chart and help the students explore the chart by asking the last five questions from the Questions for Students Questions + Reflection tab. Next, give each student a copy of the Facts I Know Activity Sheet and ask them to enter the addition facts that they know from memory.

pdficonFacts I Know Activity Sheet  

Tell the students to put their Facts I Know Activity Sheets in a safe location, because they will be using this chart in upcoming lessons.

Assessment Options

  1. The Questions for Students help the students focus on the mathematics in this lesson. They aid you in assessing the students' level of knowledge and skill. You may wish to use these questions as models for additional questions as you probe the students' understanding even further.
  2. Documenting what you discover about the students' understanding and skills on the Class Notes resource sheet provides information that can be useful when discussing the students' progress toward learning targets.


  1. Ask students to describe the visual patterns on their addition tables.  Ask them to look at diagonal line patterns, reading from the top left of the table to the bottom right, and from the bottom left to the top right.  Ask them to describe their observations.
  2. In order to further explore the Commutative Property, have students use two different color number cubes and record their addends in a table.  They will be better able to see that the two addends are different from each other using colored number cubes. 
  3. Ask students how they could use the patterns in this table to figure out an addition fact that they have forgotten.
  4. Start with a new board and invite pairs of students to take turns adding numbers of their choice.
  5. Offer students a partially completed set of cells from another addition table and ask them to complete the cells and make a conjecture about which addition table it must be.
  6. Move on to the next lesson, Finding Sums to Six.



Questions for Students 

1. What sums can you have when you roll two number cubes? What is the highest sum? The lowest?

[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12; 12; 2.]

2. What rolls of the number cubes will have a sum of 6? Of 10? Of 4?

[1+5, 2+4, 3+3; 4+6, 5+5; 1+3, 3:+2.]

3. What was the highest sum you got when you rolled the number cubes? Could you have gotten a higher sum? How do you know?

[12; No; 6 and 6 are the largest numbers on the number cubes, and 6 + 6 = 12.]

4. How can we show that 3 + 6 has the same sum as 6 + 3? What is another way? How can knowing the Commutative Property help you learn the addition facts?

[Possible answers include using a calculator, showing by counters, or counting the spots on two number cubes; if you know 6 + 5 = 11, you also know that 5 + 6 = 11.]

5. How is a hundreds chart like an addition chart? How is it different?

[Both are used to record numbers and number relationships; A hundreds chart shows the numbers 1 through 100 in order, whereas an addition chart shows sums of various pairs of numbers..]

6. What numbers will go in the far-left column? In the top row?

[0 through 9; 0 through 9.]

7. Where on the chart will you write the sign that tells that this is an addition chart?

[In the upper-left most corner is where you write the "+" symbol.]

8. What row on the addition chart will show the sums of two and another number? Which column?

[Third row; third column.]

9. Where on the chart will you find the sum of four and five? Where else?

[Where the 4 column and 5 row intersect, at 9.]

10. What do you notice about the first row of the chart? How about the first column?

[It is the same as the column heading.]

Teacher Reflection 

  • Which students know the commutative property? What experiences are necessary for those who do not?
  • Which students are able to find sums where one addend is zero, one, or two? What other sums seemed easy for them? What extension activities would be appropriate for those students?
  • Which students were able to identify the addition facts that they know at the immediate-recall level? Which students were not able to do this? What instructional experiences do they need next?
  • What adjustments will I make the next time that I teach this lesson?
Number and Operations

Finding Sums to Six

In this lesson, students discover the role of the additive identity and explore sums to six. They continue to complete their personal addition charts.
Number and Operations

Some Special Sums

Students practice doubles and doubles-plus-one addition facts. They record their current level of mastery of the addition facts on their personal addition charts.
Number and Operations

Addend Pairs to 12

Students practice their addition facts for sums up to 12 by playing a game. They add to their personal addition charts. Students are encouraged to practice the facts that they have not yet mastered. Finally, triangular flash cards help students practice addition facts.
Number and Operations

Wrapping up the Unit

In this final lesson of the unit, students display their knowledge of properties of objects for sorting and creating patterns. They also demonstrate an understanding of commutativity and model addition and subtraction of whole numbers using different representations.

Learning Objectives

Students will:

  • Find the sums of two one-digit numbers.
  • Describe patterns on an addition table.
  • Explore the commutative property of addition.
  • Record sums on an addition chart.

NCTM Standards and Expectations

  • Understand the effects of adding and subtracting whole numbers.
  • Develop and use strategies for whole-number computations, with a focus on addition and subtraction.
  • Develop fluency with basic number combinations for addition and subtraction.

Common Core State Standards – Mathematics

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.5
    Fluently add and subtract within 5.

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.B.3
    Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.B.4
    Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.C.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, Algebraic Thinking

  • CCSS.Math.Content.1.OA.D.7
    Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

Grade 1, Algebraic Thinking

  • CCSS.Math.Content.1.OA.D.8
    Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.

Grade 1, Number & Operations

  • CCSS.Math.Content.1.NBT.C.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.

Grade 2, Algebraic Thinking

  • CCSS.Math.Content.2.OA.B.2
    Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Grade 2, Number & Operations

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

Grade 2, Number & Operations

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

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.B.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.

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.B.9
    Explain why addition and subtraction strategies work, using place value and the properties of operations.

Common Core State Standards – Practice

  • CCSS.Math.Practice.MP6
    Attend to precision.