Lesson 9

This number talk encourages students to use the structure of base ten numbers to find the quotient of a base ten number and 10. The goal is to get students to see how understanding each quotient helps them find the next quotient. Reasoning about this computation will be important in both this lesson and future lessons where students are working with the metric system and percentages.

Display one problem at a time. Give students 1 minute of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their explanations for all to see. Emphasize student strategies based in place value to explain methods students may have learned about “moving the decimal point” left or right or “crossing out zeros.” To involve more students in the conversation, consider asking:

• “Who can restate ___’s reasoning in a different way?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to _____’s strategy?”
• “Do you agree or disagree? Why?”

This activity gives students first-hand experience in relating ratios of time and distance to speed. Students time one another as they move 10 meters at a constant speed—first slowly and then quickly—and then reason about the distance traveled in 1 second.

Double number lines play a key role in helping students see how time and distance relate to constant speed, allowing us to compare how quickly two objects are moving in two ways. We can look at how long it takes to move 10 meters (a shorter time needed to move 10 meters means faster movement), or at how far one travels in 1 second (a longer distance in one second means faster movement).

Along the way, students see that the language of “per” and “at this rate,” which was previously used to talk about unit price, is also relevant in the context of constant speed. They begin to use “meters per second” to express measurements of speed.

As students work, notice the different ways they use double number lines or other means to reason about distance traveled in one second.

Before class, set up 4 paths with a 1-meter warm-up zone and a 10-meter measuring zone. 

Arrange students into 4 groups, with one for each path. Provide a stopwatch. Explain that they will gather some data on the time it takes to move 10 meters. Select a student to be your partner and demonstrate the activity for the class.

• Share that the experiment involves timing how long it takes to move the distance from the start line to the finish line.
• Explain that each person in the pair will play two roles: “the mover” and “the timer.” Each mover will go twice—once slowly and once quickly—starting at the warm-up mark each time. The initial 1-meter-long stretch is there so the mover can accelerate to a constant speed before the timing begins.
• Demonstrate the timing protocol as shown in the task statement.

Stress the importance of the mover moving at a constant speed while being timed. The warm-up segment is intended to help them reach a steady speed. To encourage students to move slowly, consider asking them to move as if they are balancing something on their head or carrying a full cup of water, trying not to spill it.

Alternatively, set up one path and ask for two student volunteers to demonstrate while the rest of the class watches.

Students may have difficulty estimating the distance traveled in 1 second. Encourage them to mark the double number line to help. For example, marking 5 meters halfway between 0 and 10 and determining the elapsed time as half the recorded total may cue them to use division.

Select students to share who used different methods to reason about the distance traveled in 1 second. It may be helpful to discuss the appropriate amount of precision for their answers. Dividing the distance by the elapsed time can result in a quotient with many decimal places; however, the nature of this activity leads to reporting an approximate answer.

During the discussion, demonstrate the use of the phrase meters per second or emphasize it, if it comes up naturally in students’ explanations.  Discuss how we can use double number lines to distinguish faster movement from slower movement. If it hasn't already surfaced in discussion, help students see we can compare the time it takes to travel the same distance (in this case, 10 meters) as well as the distance traveled in the same amount of time (say, 1 second).

Explain to students that when we represent time and distance on a double number line, we are saying the object is traveling at a constant speed or a constant rate. This means that the ratios of meters traveled to seconds elapsed (or miles traveled to hours elapsed) are equivalent the entire time the object is traveling. The object does not move faster or slower at any time. The equal intervals on the double number line show this steady rate.

In the previous activity, students traveled the same distance in differing amounts of time. In this activity, students analyze a situation where two people travel for the same amount of time, each at a constant speed, but go different distances. The use of double number lines is suggested, but not required.

Monitor students’ work and notice different ways students compare speeds. For the first question, students may use meters per second or compare the distance traveled in the same number of seconds. For the last question, they may draw a double number line for each of the scenarios being compared, or calculate each speed in meters per second.

Instead of dividing 40 by 10, some students may instead calculate \(10 \div 40\). Ask them to articulate what the resulting number means (0.25 seconds to travel 1 meter) and contrast that meaning with what the problem is asking (how many meters in one second). Another approach would be to encourage them to draw a double number line and think about how they can figure out what value for distance corresponds to 1 second on the line for elapsed time.

Select students who reasoned differently to share. Some students will know that Diego ran faster, simply because he ran further, but this reasoning is not always correct. Han runs further, but is slower than Diego because he had more time. Be sure to attend to both distance and time when making the comparison. Help students draw connections between the different ways they represented and reasoned about the problem.

During the discussion, keep as much emphasis as possible on the concept of meters per second.