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Finding Fact Families

Links Away
Grace M. Burton
Location: unknown

In this lesson, students explore the relation of addition to subtraction with books and links. Then the children search for related addition and subtraction facts for a given number. They also investigate fact families, including those where one addend is 0 and where the addends are alike.

To review the concept of subtraction, read Roll Over or There Were Ten in the Bed, or sing the song Ten in the Bed from the Song Lyrics sheet.

pdficonSong Lyrics 

For each part, ask a student to act out with links what is happening, and have each student write the related subtraction sentence on a slate.

 539 fact

Now read the first three pages of Quack and Count to the students, using 6 links in one color and 1 link in another to make a chain that illustrates the text on the first three pages (for example, 7 – = 6, 7 – 6 = 1, 6 + 1 = 7, 1 + 6 = 7). If you cannot obtain the book mentioned, have a volunteer tell a story that involves subtraction by taking sets away. Model as described above. Then write the fact family.

When the students seem comfortable with this procedure, distribute links of two colors to pairs of children and have them take turns making chains, taking away the indicated group of links, and writing equations as you finish the story or as additional volunteers give subtraction problems. This activity will help the students focus on the relation of subtraction to addition. When you get to the last page, have children model 7 – 7 and write the related addition and subtraction sentences (7 – 7 = 0, 0 + 7 = 7, 7 + 0 = 7) on slates.

Then call the class together and ask students to share what they think a fact family is. Invite one of the children to make a chain with 3 links of one color and 3 of another, and write the related addition and subtraction sentences (there will be only one of each). Finally, ask the students to write a fact family for one of the chains they made. This could be added to their portfolios to document their growth in understanding.

Assessment Option

The Assessment Option is to monitor students as they write fact families on slates. Use the Class Notes recording sheet to record whether students are emerging or mastering this skill.


  1. For students who need extra support recognizing and writing fact families, place 10 two‑color chips in a cup. Have students shake up the cup and spill its contents in a shoe box lid. Then have students write the fact family illustrated by the two‑color chips. For instance, if a student spills 4 red and 2 white chips, she could write 6 – 4 = 2 and 4 + 2 = 6.
  2. Move on to the next lesson, Practice Makes Perfect.

Questions for Students 

1. What fact family can I write if I make a chain with 3 red links and 5 green links?

[I can write 8 – 3 = 5, 8 – 5 = 3, 3 + 5 = 8, 5 + 3 = 8.]

2. How are the facts in a fact family alike? How are they different?

[They are the alike because they all use the same numbers. They are different because the numbers are in different places. They are different because two equations are take away and two equations are addition.]

3. What is a fact family?

[A fact family is two addition sentences and two subtraction sentences with the same numbers. More specifically, a fact family is all the different addition and subtraction sentences you can write using three given numbers.]

4. How many addition and subtraction facts are in the fact family for a chain with 3 red and 3 blue links? Justify your answer.

[There is one addition sentence and one subtraction sentence: 6 – 3 = 3, 3 + 3 = 6]

5. How could you help a friend find a subtraction fact related to 5 + 4 = 9?

[Use 5 links of one color and 4 links of another color. Then, take away the 4 links and ask how many are left.]

Teacher Reflection 

  • Which students have some of the facts memorized?
  • Did most students remember the effects of adding or subtracting 0?
  • Which students met all the objectives of this lesson? What extension activities are appropriate for those students?
  • Are students prepared to move on or do they need more practice?
  • How can you incorporate/reinforce these skills in daily classroom routines?
  • Did students relate representations and terminology to previous experiences/lessons?

Counting Back and Counting On

This lesson, which focuses on the counting model for subtraction, begins with reading a counting book. The students model the numbers as the book is read. Then they make a chain of links and write in vertical and horizontal format the differences suggested by adding and subtracting one link at a time from their chains. Finally, they draw a chain showing one link being taken away and write in two formats the difference it represents.

Taking Away Sets

This lesson encourages students to explore another model for subtraction, the familiar set model. Reading one of the many books that feature subtraction set the stage for this lesson in which the students write story problems, find differences using sets, and present results in a table. In the discussion of the table, they focus on the effects of subtracting all and subtracting 0.

Hopping Backward on the Number Line

In this lesson, students generate differences using a number line model. Because this model highlights the measurement aspect of subtraction, it is a distinctly different representation from the models presented in the previous lessons of this unit. The order property for subtraction is investigated. At the end of the lesson, children are encouraged to predict differences and solve puzzles involving subtraction.

Finding the Balance

Pre-K-2, 6-8
This lesson encourages students to explore another model of subtraction, the balance. Students will use real and virtual balances. Students also explore recording the modeled subtraction facts in equation form.

Practice Makes Perfect

During this lesson, students use what they know about fact families to play a concentration game. They will also identify subtraction facts they need to learn.

Looking Back and Moving Forward

This final lesson reviews the work of the previous lessons and suggests a framework for summative assessment. Students will self-select a solution strategy for subtraction from the models introduced in this unit. An extension activity is suggested in which students use the mathematical knowledge and skills developed in the previous lessons to demonstrate understanding and ability to apply that knowledge to playing a new game.

Learning Objectives

Students will be able to:

  • Relate addition to subtraction.
  • Generate fact families given two addends or given one addend and the sum.

NCTM Standards and Expectations

  • Use concrete, pictorial, and verbal representations to develop an understanding of invented and conventional symbolic notations.

Common Core State Standards – Mathematics

-Kindergarten, Algebraic Thinking

  • CCSS.Math.Content.K.OA.A.1
    Represent addition and subtraction with objects, fingers, mental images, drawings1, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.

-Kindergarten, Algebraic Thinking

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

-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, 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.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.