Lesson 2

Constant Hoppers

Est. Class Sessions: 1

Developing the Lesson

Connect Repeated Addition and Multiplication. Begin this lesson by reviewing students' work with math hoppers in Unit 4 Lesson 5 Base-Ten Hoppers. Remind students that base-ten hoppers can hop forward or backwards in distances of one, ten, one hundred, and so on. Explain that in this unit, they will investigate constant hoppers, which are math hoppers that always hop a constant amount. Explain that “constant” means unchanging. For a constant hopper, the distance it jumps on each hop does not change.

Introduce the constant hoppers by using the class number line, showing a display of the number line on the first Constant Hoppers page in the Student Guide or drawing a number line on a display. Choose a number and demonstrate how a hopper would jump if it could only make jumps of that length.

Present a problem like the following:

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  • Here is a +5 (“plus five”) hopper. It always hops 5 units to the right. If it starts at 0 and hops 4 times, where will it land? (20)
  • Tell an addition sentence that describes its trip.
    (5 + 5 + 5 + 5 = 20)
  • Tell a multiplication sentence that describes its trip. (4 × 5 = 20)

Make up a few more problems about hoppers that start at 0 and hop to the right. The final resting place depends on the number of hops and the size of each hop.

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You can introduce math hoppers by making a large number line on the classroom floor, playground, or sidewalk with chalk. Students can pretend to be the math hoppers and act out the math problems.

Represent Multiplication Using Number Lines. Direct students to the Constant Hoppers pages in the Student Guide. Read the short vignette in which Professor Peabody introduces the +3 (plus 3) constant hopper. Ask students to think about the two number sentences that describe the +3 constant hopper's trip.

Ask:

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  • Look at the numbers in the addition number sentence. What does each of the numbers tell you about the +3 constant hopper's trip? (Possible responses: The threes tell you that each hop is equal to 3 units on the number line. We add four 3s together because the constant hopper hopped four times. The twelve tells you where the +3 constant hopper landed at the end of the trip.)
  • Look at the numbers in the multiplication number sentence. What do each of the numbers represent in this number sentence? (Possible responses: The four tells the number of hops the constant hopper makes. The three tells the size of each hop, and the twelve tells where the constant hopper lands at the end of the trip.)
  • Compare the two number sentences. How are they similar? (Possible responses: Both of the number sentences tell you where the constant hopper's trip ended. Both of the number sentences tell you the size of each hop.)
  • Look at both number sentences. How can you tell how many hops the constant hopper made in each sentence? (In the addition sentence you can count the number of times you add the same number. Since you are adding four 3s together that means the hopper hopped 4 times. In the multiplication sentence the number of hops is part of the number sentence.)

Ask students to work with a partner to complete Questions 1–2 in the Student Guide. In Question 2 students use their desk number line to show how the +5 constant hopper moves. Use questions similar to those asked about the +3 constant hopper to check students' understanding before assigning Questions 3–7.

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Students can use a green triangle pattern block as a hopper on their desk number lines, pointing a corner of the triangle where it lands.

Assign Questions 3–7 to students to complete independently or with a partner. After students have had time to work on these questions ask them to share their solution strategies. As students share solutions make connections between number sentences using repeated addition and those using multiplication to represent the moves for each constant hopper. For Questions 3–4 students are given the length to each hop and the number of times the constant hopper hops. Students can use their desk number line to find the solution for these questions. In Questions 5–6 students are given the length of each hop and the end point and need to determine the number of hops. Students can start on the endpoint and hop back to zero to find the number of hops.

In Questions 7A–B, students are given the number of hops and the end point and need to determine the size of the constant hopper. For Question 7A, students can use known facts and think 4 × = 20, then double the answer for Question 7B to determine the size of its hops. In Question 7C, students compare the hops made by a +5 hopper and a +10 hopper to see that the length of hops for the +10 hoppers is double the length of a hop for a +5 hopper.

Student Guide: 196 197
Answers
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