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Hopping Backward to Solve Problems

Number and Operations
Grace M. Burton
Location: unknown

In this lesson, students determine differences using the number line to compare lengths. Because this meaning is based on linear measurement, it is a distinctly different representation from the meanings presented in Lessons One and Two. At the end of the lesson, the students use reasoning and problem solving to predict differences and to answer puzzles involving subtraction.

Using chalk or masking tape, make a number line on the floor. (The students will use this to find differences on a number line by hopping from a number toward 0.) Tell the students that they will now use the number line to compare lengths. Ask one student to hop to 5 and another to hop to 3. Then ask, “Who hopped farther? How much farther?” Repeat with other students.

Next, draw a number line with the spaces one cracker apart, draw a red ring and place 3 fish-shaped crackers and a blue ring with 2 fish-shaped crackers inside. Ask: How many more fish-shaped crackers are in the ring with 3 fish-shaped crackers? How can we find out using the number line?

Diagram 1: Number line with numerals the distance of one fish apart
437 fish1 

Diagram 2: Circle with red fish and circle with blue fish
437 red fish 437 blue fish 

Encourage the students to line up the crackers from the red ring with the left end of the number line.

Then ask them to place the crackers from the blue ring in a line below the first line.

Diagram 3: Number line with fish
437 fish2 

Next show how to hop back from the end of the longer line, counting the hops aloud. Have the students record the comparison using the equation notation [3 - 2 = 1] on the Differences Activity Sheet.

pdficonDifferences Activity Sheet

It is not uncommon for the students to count the lines on the number line rather than the spaces covered by the hops. You may wish to highlight the fact that in this meaning for the operation of subtraction, spaces are counted, not points on the number line. You may demonstrate this by using a length of paper the size of a fish-shaped cracker to hop back with. After several examples, show the students that they do not need to place the crackers themselves on the number line, but can mark the length with a crayon.

To enrich the students' understanding of the number line concept, model how to use the Number Line Arithmetic Tool from the National Library of Virtual Manipulatives to compare lengths. Encourage the students to use this site during math center time, and assign students to work at the site in pairs. Those not taking their turn at the computer should complete the next activities.

Put the students into pairs and give each pair fish-shaped crackers, crayons, and one number line from the Number Lines Activity Sheet.

pdficonNumber Lines Activity Sheet

Ask each student to make two sets of crackers on a piece of paper, and then enclose each in rings of different colors. Then have the students line up the crackers carefully and draw, in the appropriate colors, a line as long as the number of crackers in the set. Then ask them to compare the lengths on the number line to find the difference and 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 numbers?") You may wish to have the students create and share similar problems. One or more of these puzzles could be added to their learning portfolios.

Assessment Option

You may find it helpful to add to your recordings on the Class Notes recording sheet you began earlier in this unit. This data may be helpful as you plan strategies for regrouping students.


  1. Pose the following challenge to students: If I have a blue pencil that is 3 inches long and a red pencil that is 5 inches long, which pencil is longer? How much longer is it? How could you use a number line to prove your answer?
  2. Move on to the next lesson, Balancing Equations.

Questions for Students 

1. How could you use the number line to compare two plates, one of which has five fish-shaped crackers and the other of which has three fish-shaped crackers?

[Draw a number line, mark the places for five and three with fish, and then compare the distance between them.]

2. What numbers have a difference of 2? Can you find some of them on the number line?

[Some examples include: 5 - 3, 4 - 2, and 3 - 1.]

3. What would be the difference if two plates had the same number of crackers on them? Would the lines that showed how many crackers are in each plate be the same length? How do you know?

[0; yes; student responses may vary.]

4. How would you explain to a friend how to compare lengths on the number line?

[Student responses may vary.]

Teacher Reflection 

  • Which students counted as they took hops, and which moved directly to the number? [The latter is an indication of a more developed number sense.]
  • 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

Counting Back

Students count back to compare plates of fish-shaped crackers, and then they record the comparison in vertical and horizontal format. They apply their skills of reasoning and problem solving during this lesson in several ways. [Because students have associated the word "more" with addition, the comparative approach to subtraction is typically more challenging for the students to understand.]
Number and Operations

How Many More?

Students write subtraction problems, model them with sets of fish-shaped crackers, and communicate their findings in words and pictures. They record differences in words and in symbols. The additive identity is reviewed in the context of comparing equal sets.
Number and Operations

Balancing Equations

This lesson encourages the students to explore another meaning for operations of subtraction, the balance. This meaning leads naturally into recording with equations. The students will imitate the action of a pan balance and record the modeled subtraction facts in equation form.
Number and Operations

Fact Family Fun

In this lesson, the relation of addition to subtraction is explored with fish-shaped crackers. The students search for related addition and subtraction facts for a given number and also investigate fact families when one addend or the difference is 0.
Number and Operations

Wrapping Up the Unit

During this final lesson in the unit, the students use the mathematical knowledge and skills developed in the previous lessons as they visit five stations to review comparative subtraction.

Learning Objectives

Students will:

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

NCTM Standards and Expectations

  • Count with understanding and recognize "how many" in sets of objects.
  • 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.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.

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.MP1
    Make sense of problems and persevere in solving them.
  • CCSS.Math.Practice.MP4
    Model with mathematics.
  • CCSS.Math.Practice.MP5
    Use appropriate tools strategically.
  • CCSS.Math.Practice.MP6
    Attend to precision.