Hopping Backward to Solve Problems

  • Hopping Backward to Solve Problems

    How Many More Fish?
    Grades: PreK to 2nd
    Periods: 1
    Author: Grace M. Burton

    Materials

    Instructional Plan

    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 fish437 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.

    Assessments and Extensions

    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.

    Extensions

    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 and Reflections

    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?
     

    Objectives and Standards

    Students will:

    • Use the number line model to find differences by comparing lengths.
    • Solve puzzles using the number line.
    Common Core State Standards – Mathematics

    Pre K to 2nd

    • Kindergarten
      • 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.

    Pre K to 2nd

    • Grade 1
      • 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.

    Pre K to 2nd

    • Grade 2
      • 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.
    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.

    Related Resources

    431icon

    Counting Back

    Number and Operations

    Grade: PreK to 2nd

    Count back to compare plates of fish-shaped crackers.
    435icon

    How Many More?

    Number and Operations

    Grade: PreK to 2nd

    Write and modeling subtraction problems.
    444icon

    Balancing Equations

    Number and Operations

    Grade: PreK to 2nd

    Use a pan balance to model and write subtraction equations.

    Fact Family Fun

    Number and Operations

    Grade: PreK to 2nd

    Relate addition to subtraction with fish-shaped crackers.
    450icon

    Wrapping Up the Unit

    Number and Operations

    Grade: PreK to 2nd

    Assess students' knowledge.