Thank you for your interest in NCTM’s Illuminations. Beginning in mid-April, all Illuminations content will be moving to Interactives will remain openly available and NCTM members will have access to all Illuminations lessons with new filtering and search options. We hope you will continue to utilize and enjoy these resources on

Pin it!
Google Plus

Using the Number Line to Compare

Number and Operations
Grace M. Burton
Location: unknown

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.

Make a chalk number line on the floor. Children will find differences on a number line by hopping from a given number toward 0. Inform the students that they will now use the number line to compare lengths. Ask one student to hop to the “5,” and another to hop to the “3.” Then ask, “Who hopped farther? How much farther?” Repeat with other students.

321 Number Jump

Next draw a number line with the spaces one cube apart and construct a train with 9 connecting cubes and another with 5 cubes. (This part of the lesson explores the meaning of “how many more?” by encouraging students to model trains and towers with different numbers of cubes.)

pdficonSample Number Line

Ask the following questions:

  • How many more connecting cubes are in the train with 9 cubes?
  • How can we find out using the number line?

Encourage the students to align the longer train with the left end of the number line. Then place the shorter train on top of it. Demonstrate how to “hop” a counter from 9 backward to 5, counting the hops aloud. Have the children record the comparison using the equation notation [9 – 5 = 4] on a copy of the Group Short and Long Trains Activity Sheet.

pdficonGroup Short and Long Trains Activity Sheet 

To forestall any misconceptions, highlight the fact that in this model, the spaces are counted, not points on the number line. After several trials, show the children that they do not need to place the trains on the number line but can mark the length with a crayon or marker that is same color as is the train.

Then put the students into pairs and give each pair connecting cubes in two colors as well as crayons and one number line. Ask each partner to make a train in a different color, draw its length on the same number line in the appropriate color, and then compare the lengths on the number line to find the difference. Ask the students to record the comparison in pictures and in equation form. After allowing time for exploration, call the students together to read their equations and share their number line illustrations.

As a concluding activity, pose puzzles such as “I am thinking of two numbers on the number line that have a difference of 5. The larger number is 6. What is the other number?” If the students are ready for a challenge, you might say only “I am thinking of two numbers on the number line that have a difference of 5. What are the two numbers?” You may wish to have the students create and share similar problems.

Assessment Options 

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.


Move on to the next lesson, Balancing.

Questions for Students 

  1. How would you use the number line to compare two trains, one of which is 5 cubes long and the other 3 cubes long?
  2. What numbers have a difference of 2? Can you find some of them on the number line?
  3. What would be the difference of two trains that were the same length?
  4. How would you explain to a friend how to compare lengths on the number line?

Teacher Reflection 

  • Which similarities did students notice?
  • Which differences did students notice?
  • What other learning experiences would enable students to compare properties of concrete objects and would also be important to the students?
  • What additional learning experiences do students need?
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

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


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:

  • Use the number line model to find differences by comparing lengths.
  • Solve and create puzzles using the number line.

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.
  • 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.

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 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, Measurement & Data

  • CCSS.Math.Content.2.MD.B.6
    Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

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.