Lesson 16

This warm-up prompts students to reason about the meaning of division by looking closely at the dividend and divisor. The expressions were purposely chosen to encourage more precise reasoning than roughly estimating. While some students may mentally solve each, encourage them to also think about the numbers in the problem without calculating. Ask them what would happen if the dividend or divisor increased or decreased. Expect students to think of fractions both as division and as numbers. Encourage connections between these two ideas.

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.

Discuss each problem one at a time with this structure:

• Ask students to indicate whether they think the expression is closer to \(\frac12\), 1, or \(1\frac12\).
• If everyone agrees on one answer, ask a few students to share their reasoning, recording it for all to see. If there is disagreement on an answer, ask students with opposing answers to explain their reasoning to come to an agreement on an answer.

In this activity, students analyze the workouts of several people on a treadmill given time-distance ratios. The purpose of this activity is to remind students how speed contexts work and to start to nudge them toward more efficient ways to compare speeds. Students see that when such ratios can be expressed with the same number of meters per minute, the ratios are equivalent and the moving objects (people, cars, etc.) have the same speed.

Speed is typically expressed as a distance per 1 unit of time, so the task provides a familiar context for computing and using rates per 1. The numbers have been chosen such that any two workouts being compared has the same time, same distance, or same speed.

Encourage students to use “per 1” and “for each” language throughout, as this language supports the development of the concept of unit rate.

As students discuss the problems, listen closely for those who use these terms as well as descriptions of speed (e.g., “same speed,” “faster,” “slower”). Also notice students who make the connections between the rates per 1 they calculated in the first half of the task and use them to answer questions in the second half. Invite some of these students or groups to share later.

Arrange students in groups of 3. Give students 2–3 minutes of quiet think time to complete the first three questions. Then, ask them to share their responses and complete the last three questions in their groups.

Specify that, when discussing the first three questions (comparisons of pairs of runners), each student in the group should take the lead on analyzing one sub-problem (i.e., sharing how the workouts of the two given runners are similar or different).

If students are not sure how to begin, suggest that they try using a table or a double number line that associates meters and minutes.

Focus the conversation on the questions in the second half of the activity, the idea of “same speed,” and the clues that two objects are moving equally fast or slow. To begin the conversation, ask: “How can you tell when things are going the same speed?” Give a moment of quiet think time before soliciting responses. Students may say: “They keep up with one another running on a track,” “same distance in the same time,” or “same miles per hour in a car,” etc.

Invite a few students to share their analyses of how the runners compare, starting with how Tyler's workout compares to Kiran's, and how Kiran's compares to Mai's. Descriptions such as “slower,” “faster,” or “higher or lower speed” should begin to emerge. After students share their analyses of Mai and Tyler's workouts, make sure to highlight that even though they ran different distances in different amounts of time, they each ran 140 meters per minute so we can say “they ran at the same speed.” This also means that Mai and Tyler's original ratios—\(4,\!200 : 30\) and \(6,\!300 : 45\)—are equivalent ratios. 

In the last problem, students need to understand that since Mai and Tyler ran at the same speed they traveled the same distance for the first 30 minutes on the treadmill. This may be difficult for students to articulate with precision, so allowing multiple students to share their thinking may be beneficial.

Students use and compare rates per 1 in a shopping context as they look for “the best deal.” The purpose of this activity is to remind students how unit price contexts work and to start to nudge them toward more efficient ways to compare unit prices.

While this task considers “the best deal” to mean having the lowest cost per unit, the phrase may have different meanings to students and should be discussed. For instance, students may bring up other considerations such as distance to store, store preference (e.g., some stores offer loyalty points), what else they need to purchase, and not wanting to buy in bulk when only a small quantity is needed. Discussing these real-life considerations, and choosing which to prioritize and which to disregard, is an important part of modeling with mathematics (MP4), but it is also appropriate to clarify that, for the purposes of this problem, we are looking for “the best deal” in the sense of the lowest cost per can.

As students work, monitor for students who use representations like double number lines or tables of equivalent ratios. These are useful for making sense of a strategy that divides the price by the number of cans to find the price per 1. Also monitor for students using more efficient strategies.

While some students may help with grocery shopping at home, it is likely many have not and will need extra information to understand what “the best deal” means.

Before students begin, ask if anyone is familiar with the weekly fliers that many stores send out to advertise special deals. Show students some advertisements from local stores, if available.

Ask students to share what “a good deal” and “the best deal” mean to them. Many students are likely to interpret these in terms of low prices (per item or otherwise) or “getting more for less money,” but some may have other practical or personal considerations. (Examples: it is not a good deal if you buy more than you can use before it goes bad. It is not a good deal if you have to travel a long distance to the store.) Acknowledge students’ perspectives and how “messy” such seemingly simple terms can be. Clarify that in this task, we are looking for “the best deal” in the sense of lowest cost per can.

5–10 minutes of quiet work time followed by whole-class discussion.

At first glance, students may look only at the number of cans in each offer or only at the price. Let students know that they need to consider the price per one can.

If any students used a representation like a double number line or table to support their reasoning, select these students to share their strategy first. Keep these representations visible. Follow with explanations from students who used more efficient strategies, and use the representations to make connections to more efficient strategies. Highlight the use of division to compute the price per can and the use of “per 1” language. The purpose of this activity and this discussion is help students see that computing and comparing the price per 1 is an efficient way to compare rates in a unit price context.