Lesson 5

More or Less

Est. Class Sessions: 2

Developing the Lesson

Part 1: Comparing and Ordering Numbers

Play Bubble Sort with 1–10. Draw a number line as in Figure 1 on the board. Remind students of the number line they made in Lesson 4.

Ask:

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  • When we used cube trains to help us make a number line, on which end of the number line did the smaller number trains go? (on the left side, nearer to the 0)
  • On which end did the larger numbers go? (more toward the right end)

Label the left end of the number line "shortest train" and the right side of the number line "longest train."

Randomly distribute the connecting cubes trains that were used in Lesson 4 to build the number line. Tell students to count the number of cubes in their trains.

Then ask:

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  • Show a partner how to count the cubes using a counting-on strategy.

Tell students they are going to play a game called Bubble Sort. Ask students with trains containing one to ten cubes to come up to the front of the room all at once. They should line up in front of the number line on the board, facing their classmates, but not in numerical order. Tell students they are going to look at their neighbors' cube trains and decide if their train has more or less cubes. See Content Note.

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The Bubble Sort. A bubble sort is a method for putting a list of items in order. For example, the items could be numbers, names, or, as in this activity, towers of cubes. The Bubble Sort compares each item of the list with the item next to it and swaps them if they are not in the correct order. The big items rise to the top like bubbles in water. There are faster sorting algorithms, but this is one of the first that computer science students study because it is simple to describe and program.

You can serve as the leader or you can choose an additional student to help lead the activity. The sorting begins when the leader says, "Compare." At the command, pairs of adjacent students compare their trains. There are two important rules for this activity: You can talk only with the two people on either side of you, and you can trade places only with a person who is next to you. The student with the smaller train moves toward the end of the number line labeled "shortest train." The student with the longer train moves toward the other end of the line. If a pair of the students are already in order, they stay in their places. Remind students that they are just comparing and switching with the student next to them until the leader says "Compare" again. Repeat the procedure until no more switches need to be made.

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As students engage in the Bubble Sort, note constructive things that they say to one another that help in the sorting. Highlight comparative communication that uses the phrases more than, less than, and about the same to assist students in their own development of mathematical communication.

Connect Representations of Numbers. Finally, the leader asks students to say the number of cubes in their trains, beginning with the student with the shortest train. As students say their numbers, label the corresponding points on the number line on the board.

Ask:

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  • What do you notice about the trains as we move from this end [near 0] to this end? (The trains get longer.)
  • Do you see a pattern? (The trains get longer by one cube at a time.)

Ask students to say the number of cubes in their trains again starting with the smallest train. As students say their numbers in order, the leader points to the corresponding number on the number line.

Ask:

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  • As the trains get longer, where does the pointer move on the number line? (The pointer moves one point at a time toward the right end of the number line.)
  • What do you notice about the numbers on the number line as the leader points to them? (The numbers get larger as the pointer moves to the right end of the number line.)

Ask students to say the numbers in order again. However, this time the student with the largest train begins the counting. The leader points to the number line as the students name their numbers from ten to one. Pose similar questions including those that make connections between the trains and the number line, and have other students demonstrate how to count the cubes in the trains.

Ask:

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  • As the trains get smaller, where does the pointer move on the number line? (The pointer moves one point at a time toward the left end of the number line.)
  • What do you notice about the numbers on the number line as the leader points to them this time? (The numbers get smaller as the pointer moves to the left end of the number line.)
  • Who has a train with more cubes than [student name]'s train? Show how to count the cubes. Find that number on the number line.
  • Who has a train with less cubes than [student name]'s train? Show how to count the cubes. Find that number on the number line.
  • Who has a train with one less cube than [student name]'s train? Show how to count the cubes. Find that number on the number line.
  • Does [student name]'s train have more or less cubes than [student name]'s train? How do you know? (Students can compare the numbers on the classroom number line or line up the trains to see which one is longer.)

Play Bubble Sort with Larger Numbers. Have the participating students return to their seats. Ask the remaining students to come to the front of the classroom with their trains. These students will have cube trains representing larger numbers. Repeat the Bubble Sort. After sorting, have students say their numbers aloud in order. Have the leader point to the numbers along the class number line.

Ask:

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  • Did the Bubble Sort work this time even though we did not use all the trains? (yes)
  • Do the trains still get longer? (yes)
  • How does the pointer move as the trains get longer? (The pointer moves along the number line toward the 130 end. The numbers get larger as the trains get longer.)
  • Whose train has more cubes, [student name]'s or [student name]'s? Show how you know.
  • Whose train has less cubes, [student name]'s or [student name]'s? How do you count the cubes?
  • Whose train has about the same number of cubes as [student name]'s?
Adventure Book:
Student Activity Book:
Master:
Number line for Bubble Sort
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