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Comparing Sets

Number and Operations
Grace M. Burton
Location: unknown

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.

pdficonAppendix A- Bibliography of Children's Counting Books

To set the stage, you may wish to read another of the counting back books listed in Appendix A. One appropriate book includes Ten, Nine, Eight, and Ten Monsters in a Bed. If you prefer, you may choose to have the students act out such subtraction situations as the following:

Call seven children to the front of the room, then roll a die to decide how many will return to their seats. Have a volunteer record on the board the subtraction equation the children acted out.

Next, model a subtraction problem in which two sets are compared. For example: Jody’s shirt has 5, buttons and Sandy’s shirt has 3 buttons.

315 Two shirts

  What questions can we answer about the shirts?

  • How many more buttons does Jody’s shirt have?
  • How many fewer buttons are on Sandy’s shirt?

Now have the students pose and answer similar questions. Then have the children write a subtraction story problem in which a set of 3 and a set of 4 are compared. Encourage them to model the problem with cubes and to share their problems with the class. Next, encourage the students to make up another problem, this time showing the comparison of a train of 5 with a set of 7. When the students are ready, call on individuals to share their problems. You may wish to suggest that they record in pictures and in equation form one of the comparing problems for their portfolios.

Display a large piece of chart paper where all the students can see it. Point out the columns you have labeled “Cubes in Shorter Train,” “Cubes in Longer Train,” and “Difference.” Have the children follow along on the Short and Long Cube Trains Activity Sheet.

pdficonShort and Long Cube Trains Activity Sheet

Display a train of 6 connecting cubes in one color and a train of 4 cubes in another color. Ask the children how the trains could be compared. Then have them dictate the entries for each column. If children have difficulty comparing, have them add cubes in a third color to the shorter train until the trains are the same length. Then suggest that the students count the cubes they added.

Provide pairs of children with connecting cubes and a piece of paper for a work mat. Tell them that they will be making trains and then comparing them. Ask them to record the number of cubes in each train and how many more cubes are in the longer train. Now have them work in pairs to create new entries for the chart. As the students identify differences, call on volunteers to enter their findings on the class chart. Allow the children time to make several entries, then call them together and review the terms “compare” and “difference.” Finally ask what would be recorded if both trains had 7 cubes. [7,7, 0] Prompt them to add other such entries.

To help the students to become more familiar with the set model for comparison subtraction, tell them you are going to teach them a game, “So Many More.” Show a train of cubes (any number greater than 1 will work.) Then roll a number cube and ask how many cubes will be on a train with that many more cubes.

315 Train-die

Make a train to verify their responses. Next ask a volunteer to start with the same number, roll the die, and make a train with that many cubes. Now compare the trains. The player with the longer train makes a tally mark. Finally, give the students a chance to play five rounds with a friend. The winner will be the child in the pair with the most tally marks.


Although the guiding questions above may assist you in understanding your children’s level of knowledge, others may suggest themselves as you watch the children at work. You may find it helpful to add to your recordings on the Class Notes Sheet that you began earlier in this unit. This data may be helpful as you plan strategies for regrouping children and for remediation or extension activities.

Questions for Students 

  1. What do we find when we compare two sets? Can you show how to compare two trains of connecting cubes?
  2. Which difference on our chart was the greatest? If we use only 10 connecting cubes, do you think we could get a larger difference? How?
  3. What would be the smallest difference we could get between two trains if one train has 10 connecting cubes? How would you get that difference?
  4. Suppose you had a train of 5 connecting cubes. How long a train would have a difference of 0 with that train? How about a difference of 5?
  5. Look at one of the rows on the chart. How would you act this out with connecting cube trains?

Teacher Reflection 

  • Which children met all the objectives of this lesson? What extension activities are appropriate for these children?
  • Which children did not meet the objectives of this lesson? What caused them particular difficulty?
  • What parts of the lesson went smoothly? Which parts would you change the next time you teach this lesson?
  • Can most of the children justify the difference when one addend is 0? Can they justify a difference of 0?
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

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:

  • Select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil.
  • Use charts to draw conclusions.
  • Become familiar with standard units.

NCTM Standards and Expectations

  • Understand the effects of adding and subtracting whole numbers.
  • 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, Counting & Cardinality

  • CCSS.Math.Content.K.CC.C.6
    Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.

-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 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.5
    Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

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.

Grade 2, Number & Operations

  • CCSS.Math.Content.2.NBT.B.9
    Explain why addition and subtraction strategies work, using place value and the properties of operations.

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.