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Counting Back and Counting On

Number and OperationsAlgebra
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.

To open this lesson, choose a counting book, such as one of those listed in the Bibliography of Children's Counting Books. 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.” 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.


1. 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.

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

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.