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

Where Will I Land?

Number and Operations
Location: Unknown

In this lesson, the students find differences using the number line, a continuous model for subtraction. [Number can be considered in two ways: discrete and continuous. The counting and set models use the discrete form of number.] Students are encouraged to predict differences and to compose puzzles involving subtraction.

Note: Before the lesson begins, attach a long strip of masking tape to the floor and draw a number line on it. If you prefer, you might draw a chalk number line on the floor. Label the line from 0 to 12.

Inform the students that today they will use a number line to find differences. Review addition on the number line by presenting an addition sentence such as 5 + 4 = __, and have volunteers show how to hop on the large number line to find the sum.

683 frogline

Then display a subtraction example such as 8 – 3 = __, and call on a volunteer to tell a number story that would fit that subtraction situation. Then ask: How can we find the difference using the number line? Guide a volunteer to stand on the 8 and "hop" back three spaces on the number line. Ask the volunteer to tell his or her present position. [5] If this model is new to the students, you may wish to encourage them to count backward as each hop is made. Ask for volunteers to record what happened on the number line using both the vertical and the horizontal format of the equation notation. Then encourage them to tell a subtraction story that describes the moves for 9 - 5. [If you start at 9 and take 5 hops backward, you land on 4.] Remind the students that, spaces, not points, are counted in operations on the number line.

After the class has seen several examples, place the students in pairs and give each pair some pasta shapes, a number cube, a set of index cards numbered to 10, and a strip of masking tape to write numbers on to use as a number line. Or, you may distribute individual number lines.

pdficonNumber Lines

Ask the students to take turns with one student showing an index card and rolling a number cube and the second student making up a subtraction story with the numbers from the card and the number cube. Then have the second student move a pasta shape on the number line to find the difference of the smaller number from the larger number. Ask the first student to record the hops in pictures related to the story and in two other forms. Then have the students switch roles. Encourage the students to predict the differences before they verify their predictions by moving a pasta shape on the number line. One or more of these puzzles and their solutions could be added to the students' learning portfolios.

683 mac math4 

When the pairs have finished, call the class together to share some of the problems they wrote and tell how they found and recorded the differences. Then pose this problem:

I am the number you land on when you start at 4 and hop back 2--what am I?

Ask for volunteers to create similar problems and other volunteers to find their answers by using the large number line.

  • Masking tape
  • Markers
  • Paper
  • Crayons
  • Index cards
  • Number cubes
  • Number Lines


Assessment Options

  1. The Questions for Students will help the students focus on the mathematics in this lesson. They will also aid you in understanding the students' current level of knowledge and skill with the mathematical concepts presented.
  2. You may wish to document your observations about student understandings and skills on the Teacher Resource Sheet, Class Notes, begun earlier in this unit plan. These comments may be useful when you are planning additional learning experiences for individual students.
Move on to the next lesson, What Balances?

Questions for Students 

1. What number will you land on if you start at 10, then hop back 3? Can you demonstrate that on the number line? Can you draw a picture of what you did?

[7; 10 - 3 = 7.]

2. What differences did you model with hops when you worked in pairs? How did you record them?

[Answers will depend upon the examples used by the students.]

3. Were you able to predict any of the differences? Which ones?

[Answers will depend upon the examples used by the students.]

4. Were any of the differences the same? Which ones?

[Answers will depend upon the examples used by the students.]

5. How will you find the difference of 8 and 5? How will you record that? Can you record it any other ways?

[8 - 5 = 3; answers may vary.]

6. Suppose you started at 5 and landed on 3. How many spaces did you hop back? Can you show us that on the number line?

[2 spaces; 5 - 3 = 2.]

7. Suppose you started at 5 and hopped back 5. Where would you land? Suppose you started at 5 and hopped back 0. Where would you land?

[0; 5.]

8. How could you tell a friend to subtract using the number line?

[Students should be able to explain how to subtract using the number line.]

Teacher Reflection 

  • Which students are eager to volunteer? Which students do not volunteer even when they know the answer? How can I encourage them to share what they know?
  • Which students counted as they took hops, and which moved directly to the number?
  • Which students are comfortable using this learning tool?
  • What activities would be appropriate for students who met all the objectives?
  • Which students had trouble using the number line? What instructional experiences do they need next?
  • What adjustments will I make the next time that I teach this lesson?
Number and Operations

Recording Two Ways

The students make sets of pasta shapes and count some away, then record the subtraction in vertical and horizontal formats. They draw a set and cross out some shapes, then write in both formats the subtraction that the drawing represents.
Number and Operations

How Many Left?

This lesson encourages the students to explore the familiar set model of subtraction. The students write story problems and find differences using sets, including subtracting all and subtracting zero. They record the differences in a chart.
Number and Operations

What Balances?

This lesson encourages students to explore another meaning of subtraction, the balance. They use subtraction facts to generate related addition facts and explore at the concrete level the idea of subtraction as the inverse of addition.
Number and Operations

Who's in the Fact Family?

In this lesson, the exploration of the relation of addition to subtraction is continued as the students use problem-solving skills to find fact families, including those in which one addend is zero or in which the addends are alike.
Number and Operations

What's the Difference?

During this lesson, students use reasoning to find differences from numbers up to 10, using real and virtual calculators and an addition chart as tools. They also play a concentration game.

Learning Objectives

Students will:

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

NCTM Standards and Expectations

  • 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 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, Counting & Cardinality

  • CCSS.Math.Content.K.CC.A.3
    Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

-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.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, 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.