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Counting Back and Counting On

Links Away
Grace M. Burton
Location: unknown

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.

To set the stage for learning, choose a counting back book to read from the Counting Books Resource Sheet, such as Ten Sly Piranhas, Ten Little Ladybugs, or Five Little Monkeys Sitting in a Tree.

pdficonCounting Books 

If no books are available, tell a story about counting back such as the following:

There were eight candies on a plate. Zack took one of them; then Yula took one; then Xavier took one, and Wilma took one, and so forth.
As the students listen to the story, have them model each number from the story with counters, links, or candies, then write the numeral and number word for each. Watching their responses will help you assess the students’ current level of understanding.
524 chain links

Next give each child 10‑12 links. Hold up a number and ask the children to make a chain with that many links, lay it flat on the desk and then record the number of links on their slates. Then ask them to add one link and record the addition they have just modeled using the equation (horizontal) format. Now show the same starting number and ask the children to make a chain with that many links. Have students record that number as before but this time to count away one link and then record the subtraction they have just modeled using the horizontal format.

524 held links

Now ask the students to build a chain of six links and hold the chain vertically so that it hangs down. Lead them in singing Six Little Ducks, from the Song Lyrics Resource Sheet, as they remove the links one at a time. Ask them to record each subtraction in vertical format.

pdficonSong Lyrics

When the children are ready, ask them to share one of their differences and the counting process that was used to get it. Summarize how students can count back to subtract one.

Assessment Options

  1. Document your observations during class discussions and student practice. Use the Class Notes recording sheet.
  2. Ask students to choose one subtraction example and write it in both vertical and horizontal format for an entry in their portfolio. Have them draw a picture that illustrates the subtraction fact.


  1. Put several counting back books in a math center along with paper, crayons and links. This availability will encourage children to continue to read books with a subtraction theme and to practice recording subtraction examples.
  2. Move on to the next lesson, Taking Away Sets.

Questions for Students 

1. How many links are in this chain (show a chain with ten links)? On this chain (show a chain with nine links)? Which chain has more? How many more?

[The chain with none links has more. It has one more link than the chain with ten links.]

2. What number sentence would show that you started with 10 links and took one away?

[10 – 1 = 9.]

3. What was alike between the two ways we recorded subtraction? What was different?

[The numbers we used were the same. We got the same answer. They look different because one is going across and one is going up and down.]

4. How could you help a younger child model this subtraction sentence (write 10 – 1)?

[Tell her to start with ten things and take one away.]

5. What does the minus sign mean?

[It means "take away."]

6. What two symbols did you use to mean equals?

[Two short lines and one long line. Students may point to the two symbols.]

7. When you count backwards, what comes after seven?

[Six comes after seven.]

Teacher Reflection 

  • Which books were best received? What others might you use?
  • Were the materials used for counting back appropriate?
  • Which students met all the objectives of this lesson? What extension activities would be appropriate for those children?
  • Which students did not meet the objectives of this lesson? What instructional experiences do they need next?
  • Do students understand the symbols for subtract and equals?
  • What mathematical ideas need clarification?

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.

Finding Fact Families

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.

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:

  • Count to 10.
  • Model numbers to 10.
  • Count differences from numbers to 10.
  • Write and recognize numerals to 10.
  • Record differences in vertical and in horizontal format.

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.A.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.

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 2, Algebraic Thinking

  • CCSS.Math.Content.2.OA.A.1
    Use addition and subtraction within 100 to solve one- and two-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.

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.