Lesson 6

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?”

In this task, students practice finding unit prices, using different reasoning strategies, and articulating their reasoning. They also learn about the term “at this rate.”

As students work, observe their work and then assign one problem for each group to own and present to the class. (The problems can each be assigned to more than one group). Have them work together to create a visual display of their problem and its solution.

Adjust this activity to 10 minutes.

In the launch, adjust the displayed problem to: “Pizza costs $10 for 8 slices. At this rate, how much will 6 slices cost?” During the discussion of the phrase “at this rate,” ask students to outline a method for solving the problem. Highlight any methods that determine the cost for 1 slice of pizza. Tell students that the price for 1 item is called the unit price and can be described using the word “per.” For example, “This pizza costs $1.25 per slice.”

Tell students to omit the last question in which they create a visual display.

The first and third questions involve using decimals to represent cents. If the decimal point is forgotten, remind students that the cost of the bracelet is less than one dollar, and the cost of the chips is in between one and two dollars.

Watch for students working in cents instead of dollars for the bracelets. They may come up with an answer of 275 cents. For these students, writing 25 cents as \$0.25 should help, or consider reminding them of the avocados from a previous activity, which had a unit price of \$0.50.

Invite each group to present its assigned problem. After each group presents, highlight the group’s strategy, accurate uses of the terms “at this rate” and “per,” and the ways in which a double number line might have been used when working with unit price.

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.