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Fact Families

Number and Operations
Grace M. Burton
Location: unknown

In this lesson, the relationship of addition to subtraction is explored with books and with connecting cubes. Students search for related addition and subtraction facts for a given number using a virtual or actual calculator to find differences. They also investigate fact families when one addend is 0 as well as when the addends are the same.

To review the concept of subtraction, read Ten Monsters in the Bed, or sing the song “Ten in the Bed.” Call on volunteers to demonstrate with connecting cubes what is happening and to write or draw pictures of the related addition and subtraction sentences for each part—for example, 10 – 1 = 9 and 9 + 1 = 10.

Then call out a difference (for instance, “5”) and have each student show you the meaning of the subtraction sentence by making two connecting cube trains that have the stated difference. Examples include trains of 7 and 2, 3 and 8, and 4 and 9. Next have the students add cubes in a third color to the shorter train and describe the relationship between the trains in as many ways as they can using subtraction sentences and then related addition sentences. [Examples include 7 – 2 = 5, 7 – 5 = 2, 5 + 2 = 7, and 2 + 5 = 7.] When the children seem comfortable with this exercise, distribute connecting cubes in three colors to pairs of children and have them take turns making trains and writing equations. Encourage the students to make two trains, each 7 cubes long, and write the related addition and subtraction sentences: 7 – 7 = 0 and 0 + 7 = 7.

Then call the class together and ask a volunteer to make two trains and write the four number sentences (two addition and two subtraction) that the trains suggest. You may wish to repeat this with other volunteers. Invite one of the students to make a train with three connecting cubes of one color and three of another and write the related addition and subtraction sentences. Note that there will be only one addition and one subtraction sentence.

329-L41 Train

Finally, ask the students to write a set of number sentences about one of the trains that they made.

Display the online Calculators and Hundred Boards: Displaying Number Patterns tool and allow groups of students to take turns finding differences using the online calculator.

appicon Calculators and Hundred Boards: Displaying Number Patterns 

In addition, children can use actual calculators to find several pairs of numbers less than 12 that have a difference of “2” and then record the subtraction sentences. Ask students to repeat the exercise with another difference, such as “3.” Ask students to explain the number sentences, how and why they are alike and different.

pdficonFact Families Activity Sheet 

Students should record the fact families they have identified in this lesson on the Fact Families Activity Sheet.

Assessment Options 

  1. The children’s responses to the guiding questions can help you understand their current level of understanding. After this lesson, you may wish to add more documentation to the Class Notes recording sheet. These notes will be valuable as you plan appropriate remediation and enrichment opportunities.
  2. Collect the Fact Families Activity Sheet and use this to assess students' knowledge of the various fact families identified in this lesson.


  1. Challenge Question: How would you explain to a classmate how to find all the members of a fact family?
  2. Move on to the last lesson, Looking Back and Moving Forward.

Questions for Students 

  1. If you know one train has 7 connecting cubes and another has 2 cubes, how many more cubes are in the longer train?
  2. How many addition and subtraction facts can I write if I compare a train with 3 red connecting cubes and one with 5 green connecting cubes? How are the facts alike? How are they different?
  3. Suppose that I make a train with 4 red connecting cubes and one with 4 blue connecting cubes. What sentences will describe a comparison of the trains?
  4. How could you help a friend find an addition sentence related to 5 – 2 = 3? To 4 – 0 = 4?
  5. What addend pairs can you find for a sum of 7? What subtraction sentences do they suggest?
  6. Suppose I tell you that 7 is 3 more than some number. Can you write the complete subtraction sentence? If 7 were less than a number, what sentence would you write?
  7. If I have a blue pencil that is 3 inches long and a red pencil that is 5 inches long, how can I find out how much longer the red pencil is?

Teacher Reflection 

  • Did most children remember the effects of adding or subtracting 0?
  • Which children met all the objectives of this lesson? What extension activities are appropriate for those children?
  • Which children are still having difficulty with the objectives of this lesson? What additional instructional experiences do they need?
  • Which children are most dependent upon the manipulatives? Which children are beginning to find the answers without them?
  • What will you do differently the next time that you teach this lesson?
Number and Operations

Counting Back and Counting On

In this lesson, students model subtraction with connecting cubes while the teacher reads to them from counting books. Then children make a train of connecting cubes and write in vertical and horizontal format the differences suggested by adding to and subtracting from the train one cube at a time. Finally, they record, in written form, a train showing one cube being taken away and record the difference in vertical and horizontal format.
Number and Operations

Comparing Sets

A children’s book sets the stage for this lesson which encourages students to review counting back. In this lesson, children write subtraction problems and model them with cubes. They compare sets and record differences in the form of a table. The additive identity is reviewed in the context of comparing equal sets.
Number and Operations

Using the Number Line to Compare

In this lesson, students determine differences using the number line to compare lengths. Because this model is based on linear measurement, it is a distinctly different representation from the models presented in the previous two lessons. At the end of this lesson, children are encouraged to predict differences and answer puzzles involving subtraction.
Number and Operations


This lesson encourages students to explore another model of subtraction, the balance. This model leads naturally to recording with equations. Students use actual and virtual pan balances in their explorations and record the modeled subtraction facts and the related addition facts in equation form.
Number and Operations

Looking Back and Moving Forward

This final lesson of the unit reviews the work of the previous lessons and suggests a framework for summative assessment. During this lesson, students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and the ability to apply that knowledge to playing subtraction games.

Learning Objectives

Students will:
  • Find missing addends.
  • Review the additive identity.
  • Generate fact families given two addends or given one addend and the sum.

NCTM Standards and Expectations

  • Count with understanding and recognize "how many" in sets of objects.
  • Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
  • Use multiple models to develop initial understandings of place value and the base-ten number system
  • Understand the effects of adding and subtracting whole numbers.
  • Understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations.
  • Use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.

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.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.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.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.

Common Core State Standards – Practice

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