Lesson 8

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying and dividing fractions. These understandings help students develop fluency and will be helpful throughout this unit when students find constants of proportionality from graphs, tables, and equations. While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem, saving most of the time for the final question.

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their answers and explanations for all to see.

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?”

If time permits, ask students if they notice any connections between the problems. Have them share any relationships they notice.

In this activity, students graph three time-distance relationships along with the one from the previous lesson, “Tyler’s Walk.” One of these is not a proportional relationship, so students must pay close attention to the quantities represented. The purpose of this activity is to give students many opportunities to connect the different features of a graph with parts of the situation it represents. In particular, they attach meaning to any point that is on a graph, and they interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second. Comparing different but related situations and their graphs supports students as they make sense of the situation.

Students may expect the graphs to intersect because everyone arrives at the same location. However, they did not arrive there at the same time (with the exception of Tyler and Mai). Because all characters traveled the same distance from the ticket booth and no further, the endpoints of their graphs lie on the same horizontal line \(y = 50\), that is, they have the same \(y\)-coordinate. The points will vary in position from right to left depending on the number of seconds after Tyler left the ticket booth it took each person to arrive at the bumper cars. Note this features in a whole-class discussion.

The most important goals of the discussion are to attach meaning to any point that is on a graph, and to interpret the meaning of the distance when the time is 1 second as both the constant of proportionality of the relationship and the person’s speed in the context in meters per second.

These questions may be used to facilitate the class discussion:

• "For each graph that shows a proportional relationship, what is the constant of proportionality?" (Tyler's was 1.25, Diego’s was \(1\frac23\), Lin’s was 2.5.)
• "How did you find them?" (Answers will vary, but students could have divided a \(y\)-coordinate by its associated \(x\)-coordinate.)
• "Where do constants of proportionality occur in the tables, and where do they occur on the graphs?" (They occur in the tables as the values in the second column that correspond to the value of 1 in the first column; on the graphs as the \(y\)-coordinate of points where the \(x\)-coordinate is 1.)
• "Which is the only graph that does not represent a proportional relationship?"
• "A classmate argues that Mai's graph must represent a proportional relationship, because she jogged at a steady rate. How do you answer?" (Mai's graph does not pass through the origin, so it does not represent a proportional relationship. That is, the distance she traveled is not proportional to the time elapsed. What does “time elapsed” mean? It is time elapsed since Tyler left the ticket booth. However, distance vs. time elapsed since Mai left the ticket booth until she arrived at the bumper cars is a proportional relationship.)

This activity is intended to help students interpret features of graphs in terms of the proportional relationships they represent. This task highlights two fundamental ideas:

• The steeper the graph, the greater the constant of proportionality.
• If the graph represents distance of an object vs. time, the constant of proportionality is the speed of the object.

Students can reason abstractly (MP2) by picking an arbitrary time and comparing the corresponding distances, or picking an arbitrary distance and comparing the corresponding times. Ideally, both of these ways of reasoning are shared in a whole-class discussion of the task because they will be needed in future work.

While students work, monitor for these approaches:

• At the same time, Asteroid x has traveled a greater distance (as in the graph with the vertical dashed line). We can’t tell exactly where 1 unit of time is on the graph, but wherever it is, we can tell that Asteroid x has covered a greater distance.

• Asteroid x takes less time than Perseid 245 to cover the same distance. See the graph with the horizontal dashed line.
• When distance is proportional to time and distance is graphed against time, the constant of proportionality represents the magnitude of the speed (unit of distance traveled per unit of time). It was shown in the previous activity that a steeper line has a greater constant of proportionality. Therefore, a line steeper than Perseid 245’s line represents a greater speed.

It is important that students not assume “steeper always means faster,” but that they understand why it is in this case by reasoning abstractly and attending to the referents for points on the graphs. If the same relationships were graphed with distance on the horizontal axis and time on the vertical axis, a steeper line would indicate a slower speed. If the same relationships were graphed on separate axes, their scales could be different. Because the graphs share the same axes, it is implicit that comparisons between them occur relative to the same units.

Try to find students who took each approach, and invite each to share reasoning with the class. Important points to highlight are:

• A steeper graph has a larger constant of proportionality and a larger constant of proportionality will have a steeper graph. (“When you look at two graphs of a proportional relationship, how can you tell which one has a greater constant of proportionality?”)
• In a distance vs. time graph, a steeper graph indicates a greater speed. (“When you look at two distance vs. time graphs, how can you tell which represents an object traveling at a greater speed?”)
• The two graphs in question 1 can be considered from two perspectives:
  • Same point in time, noting that one object has covered a greater distance. (“Which student focused on the two objects at the same moment in time? How did they know that Asteroid x was traveling faster?”)
  • Same distance, noting that one object took less time. (“The other student did not focus on the same moment in time. What did they focus on in her explanation?”)