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Counting Back and Counting On

Number and Operations
Grace M. Burton
Location: unknown

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.

pdficonAppendix A- Bibliography of Children's Counting Book

To open this lesson, choose a counting book, such as one of those listed in Appendix A. One Gorilla or the M&M’s Counting Book present the same content in very different ways. As children listen to the story, have them model each number they hear by adding one connecting cube to a train of cubes and writing the numeral and number word for each model.

310-4 Block Train

(Note that if you would like a more permanent record of their present level of understanding, have them color graph-paper squares to show the number. If you choose not to read a counting book, you might sing a familiar counting song, such as “1, 2, 3, 4, 5, I caught a fish alive.”)

Next, give each child ten to twelve connecting cubes. Hold up a number, such as 8, and ask the children to make a train with that many cubes, lay it flat on the desk, and record the number of cubes. Then ask students to make a second train with one less cube and then put the trains together to compare their lengths. Ask them to record the comparison they have just modeled using the equation, or horizontal, format—for example, 8 – 7 = 1. Now have students record the comparison using the vertical format.

Next, chose a “counting back” book, such as Ten Sly Piranhas, Ten Monsters in the Bed, or Five Little Monkeys Jumping on the Bed. Have children take one connecting cube at a time away from a cube train as the story is read and record each subtraction in equation form. Then have them stack ten cubes vertically in a tower. Ask the students to remove one cube at a time as you count backwards from ten or reread the story. Ask them to record each subtraction in the vertical format.

When the students are ready, name a number and ask then to tell or show with cubes what number is “one more” and which is “one less.”

pdficonOne More, One Less Activity Sheet

You may wish to record their responses on the One More, One Less Activity Sheet. Ask the children to draw trains that show one more and one less. Then ask them to repeat the activity with towers. You may wish to ask them to save these as entries for their portfolio.

Assessment Option

Documenting information about students’ understanding and skills throughout the unit can help you focus on individual needs and strengths and foster appropriate additional learning opportunities. A recording sheet, Class Notes, is provided. You may find the information you record useful when discussing children’s progress toward learning targets.


  1. Put several “counting back” books from the Bibliography of Children's Counting Books (or other sources) in a center along with paper, crayons, and connecting cubes. These items will encourage children to continue to read books with a subtraction theme and to practice recording subtraction examples.
  2. Continue on to the next lesson, Comparing Sets.

Questions for Students 

1. How many connecting cubes are in this train? (Show a train with ten connecting cubes.) On this train? (Show a train with nine connecting cubes.) Which train has more? How many more? Which train has less? How many less?

[10; 9; the one with 10 has 1 more; the one with 9 has 1 less.]

2. What number sentence would show that you started with eight connecting cubes and compared it with a train with nine cubes? One with ten cubes?

[9 - 8 = 1; 10 - 8 = 2.]

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

[They both have the same numbers, they are just displayed differently.]

4. How could you help a younger child model 7 – 1?

[Student responses may vary.]

5. Can you write an equation to show that you compared a train of eight connecting cubes with a train with seven cubes?

[8 - 7 = 1.]

6. What does the minus sign mean?

[It means to subtract.]

Teacher Reflection

  • Were the books you chose well received? What others might you use?
  • Which children met all the objectives of this lesson? What extension activities would be appropriate for those children?
  • Which children did not meet the objectives of this lesson? What instructional experiences do they need next? What mathematical ideas need clarification?
  • Were all students able to model the numbers?
  • What adjustments would you make the next time that you teach this lesson?
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

Fact Families

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.
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:

  • Count to 10.
  • Model numbers to 10.
  • Write and recognize numerals to 10.
  • Subtract 1 from numbers to 10.
  • Record differences in vertical and in horizontal format.

NCTM Standards and Expectations

  • Develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers.
  • Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections.
  • 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.
  • 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.
  • 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.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.5
    Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

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.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.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP6
    Attend to precision.