Lesson 14

This activity reminds students of previous work they have done with constant speed situations, using \(d=rt\) for the relationship between distance, rate, and time. This prepares students for representing movement in opposite directions using signed numbers in the rest of this lesson.

Students may fall back to earlier methods to make sense of these problems and come up with a solution, like creating a double number line or a table of equivalent ratios relating distance and time. These approaches are fine. In the discussion, though, ensure everyone understands using \(d=rt\) to represent the relationship between distance traveled, elapsed time, and rate of travel for constant speed situations. 

Ask students what they remember about problems involving distance, rate, and time. They might offer that distances traveled and elapsed time creates a set of equivalent ratios, or that the elapsed time can be multiplied by the speed to give the distance traveled. Give students 1 minute of quiet work time followed by whole-class discussion.

Some students may struggle knowing whether they should multiply or divide the numbers in each problem situation. Remind them of the equation \(d=rt\).

The most important thing for students to remember is that the equation \(d=rt\) can be used to solve problems involving movement at a constant speed.

• To find the distance traveled, you can multiply the rate of travel (or speed) by the elapsed time.
• To find the rate of travel (or speed), you can divide the distance by the elapsed time.
• To find the elapsed time, you can divide the distance traveled by the rate of travel (or speed).

Consider drawing a table to facilitate the discussion of each problem and to remind students of the strategies they used while working with proportional relationships, such as using a scale factor or calculating the constant of proportionality. When relating distance and time in a constant speed situation, the speed is the constant of proportionality.

The purpose of this activity is for students to encounter a concrete situation where multiplying two positive numbers results in a positive number, and multiplying a positive and a negative number results in a negative number. 

Students use their earlier understanding of a chosen zero point, location relative to this as a positive or negative quantity and description of movement left (negative) or right (positive) along the number line, with different speeds. They extend their understanding to movement with positive and negative velocity and different times. This will produce negative or positive end points depending on if the velocity is negative or positive. Looking at a number of different examples will help students describe rules for identifying the sign of the product of a negative number with a positive number (MP8).

Display the image to remind students of west (left, negative) and east (right, positive) from the previous activity. Describe that we can talk about speed in a direction by calling it velocity and using a sign, so negative velocities represent movement west, and positive velocities represent movement east.

Encourage students who get stuck to use the provided number line to represent each situation.

The most important thing for students to understand from this activity is that if we multiply two positive numbers the result is positive and that if we multiply a positive and a negative number the result is negative.

Ask a student to share their rationale about each problem. Display the number line from the launch, and place the negative answers in the context of the problem (to the west). Make sure the distinction is made between the velocity (the direction of movement) and the position. Then, ensure students see that at least in this case, it appears that when we multiply two positive values, the product is positive. But when we multiply a positive by a negative value, the product is negative. We are going to take this to be true moving forward, even if the numbers represent other things.

In this lesson, students will interpret negative time in context. The warm-up primes them for those interpretations.

Arrange students in groups of 2. Give students 30 seconds of quiet think time, followed by partner discussion.

Ask students to come to agreement with their partners, and help them to productively resolve any discrepancies. Point out that if she is walking at a constant speed, then her positions before and after will be equally far from her position in the picture.

Students use their earlier understanding of a chosen zero point and description of positive and negative velocity, and extend this to include negative values for time to represent a time before the time assigned chosen as zero. This will produce different end points depending on if the velocity or time is negative or positive.  Students use the context to help make sense of the arithmetic problems (MP2). Looking at a number of different examples will help students describe rules for identifying the sign of the product of two negative numbers (MP8). Students may choose to use a number line to help them in their reasoning; this is an example of using appropriate tools strategically (MP5).

Keep students in the same groups. Remind the students of movement east or west as positive or negative velocity.

This activity is the same context as one in the previous lesson, and the questions are related. So students should be able to get to work rather quickly. However, each question requires some careful thought, and one question builds on the other. Consider suggesting that students check in with their partner frequently and explain their thinking. Additionally, you might consider asking students to pause after each question for a quick whole-class discussion before continuing to the next question.

If students struggle to calculate the velocity, ask them how fast is the car going after 1 second.

The key thing for students to understand here is that a negative multiplied by another negative is a positive. The last two rows in the table and the final two questions are the keys to this so draw attention to the logical progression that movement in the negative direction will have a positive position when time is negative.