Lesson 14
Position, Speed, Direction
14.1: Distance, Rate, Time (5 minutes)
Warmup
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.
Launch
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 wholeclass discussion.
Student Facing
 An airplane moves at a constant speed of 120 miles per hour for 3 hours. How far does it go?
 A train moves at constant speed and travels 6 miles in 4 minutes. What is its speed in miles per minute?
 A car moves at a constant speed of 50 miles per hour. How long does it take the car to go 200 miles?
Student Response
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Anticipated Misconceptions
Some students may struggle knowing whether they should multiply or divide the numbers in each problem situation. Remind them of the equation \(d=rt\).
Activity Synthesis
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.
14.2: Velocity (15 minutes)
Activity
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).
Launch
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.
Supports accessibility for: Conceptual processing; Visualspatial processing
Student Facing
A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.
Vehicles that are traveling towards the east have a positive velocity, and vehicles that are traveling towards the west have a negative velocity.
 Complete the table with the position of each vehicle if the vehicle is traveling at a constant speed for the indicated time period. Then write an equation.
velocity
(meters per
second)time after passing
the camera
(seconds)ending
position
(meters)equation
describing
the position+25 +10 +250 \(25 \boldcdot 10 = 250\) 20 +30 +32 +40 35 +20 +28 0  If a car is traveling east when it passes the camera, will its position be positive or negative 60 seconds after it passes the camera? If we multiply two positive numbers, is the result positive or negative?
 If a car is traveling west when it passes the camera, will its position be positive or negative 60 seconds after it passes the camera? If we multiply a negative and a positive number, is the result positive or negative?
Student Response
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Student Facing
Are you ready for more?
In many contexts we can interpret negative rates as "rates in the opposite direction." For example, a car that is traveling 35 miles per hour is traveling in the opposite direction of a car that is traveling 40 miles per hour.
 What could it mean if we say that water is flowing at a rate of 5 gallons per minute?

Make up another situation with a negative rate, and explain what it could mean.
Student Response
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Anticipated Misconceptions
Encourage students who get stuck to use the provided number line to represent each situation.
Activity Synthesis
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.
Design Principle(s): Support sensemaking; Optimize output (for generalization)
14.3: Before and After (5 minutes)
Warmup
In this lesson, students will interpret negative time in context. The warmup primes them for those interpretations.
Launch
Arrange students in groups of 2. Give students 30 seconds of quiet think time, followed by partner discussion.
Student Facing
Where was the girl:
 5 seconds after this picture was taken? Mark her approximate location on the picture.
 5 seconds before this picture was taken? Mark her approximate location on the picture.
Student Response
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Activity Synthesis
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.
14.4: Backwards in Time (15 minutes)
Activity
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).
Launch
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 wholeclass discussion before continuing to the next question.
Supports accessibility for: Organization; Attention
Design Principle(s): Cultivate conversation; Support sensemaking
Student Facing
A traffic safety engineer was studying travel patterns along a highway. She set up a camera and recorded the speed and direction of cars and trucks that passed by the camera. Positions to the east of the camera are positive, and to the west are negative.

Here are some positions and times for one car:
position (feet) 180 120 60 0 60 120 time (seconds) 3 2 1 0 1 2  In what direction is this car traveling?
 What is its velocity?


What does it mean when the time is zero?

What could it mean to have a negative time?


Here are the positions and times for a different car whose velocity is 50 feet per second:
position (feet) 0 50 100 time (seconds) 3 2 1 0 1 2  Complete the table with the rest of the positions.
 In what direction is this car traveling? Explain how you know.

Complete the table for several different cars passing the camera.
velocity
(meters per
second)time after passing
the camera
(seconds)ending
position
(meters)equation car C +25 +10 +250 \(25\boldcdot 10 = 250\) car D 20 +30 car E +32 40 car F 35 20 car G 15 8 
 If a car is traveling east when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?

If we multiply a positive number and a negative number, is the result positive or negative?

 If a car is traveling west when it passes the camera, will its position be positive or negative 60 seconds before it passes the camera?
 If we multiply two negative numbers, is the result positive or negative?
Student Response
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Anticipated Misconceptions
If students struggle to calculate the velocity, ask them how fast is the car going after 1 second.
Activity Synthesis
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.
Lesson Synthesis
Lesson Synthesis
Key takeaways:
 We can choose a zero point and then positive and negative numbers can represent positions to the right or left of this zero point.
 Signed numbers can also be used to represent speed in opposite directions. This is called velocity.
 Positive times are after a chosen zero time, and negative times are times before the chosen zero time.
 A positive times a positive is always positive.
 A negative times a positive or a positive times a negative is always negative.
 A negative times a negative is always positive.
Discussion questions:
 How can we represent a position to the left or right of a starting point without using direction words?
 How can we represent how fast something is moving to the left or right from a starting point? What word do we use to represent speed with a direction?
 How can we represent a time that came before a specific zero point?
 What kind of number do you get when you multiply a negative number by a positive number? Use a context from the lesson to explain why this makes sense.
 What kind of number do you get when you multiply a negative number by a negative number? Use a context from the lesson to explain why this makes sense.
14.5: Cooldown  True Statements (5 minutes)
CoolDown
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Student Lesson Summary
Student Facing
We can use signed numbers to represent the position of an object along a line. We pick a point to be the reference point, and call it zero. Positions to the right of zero are positive. Positions to the left of zero are negative.
When we combine speed with direction indicated by the sign of the number, it is called velocity. For example, if you are moving 5 meters per second to the right, then your velocity is +5 meters per second. If you are moving 5 meters per second to the left, then your velocity is 5 meters per second.
If you start at zero and move 5 meters per second for 10 seconds, you will be \(5\boldcdot 10= 50\) meters to the right of zero. In other words, \( 5\boldcdot 10 = 50\).
If you start at zero and move 5 meters per second for 10 seconds, you will be \(5\boldcdot 10= 50\) meters to the left of zero. In other words,
\(\displaystyle \text5\boldcdot 10 = \text50\)
We can also use signed numbers to represent time relative to a chosen point in time. We can think of this as starting a stopwatch. The positive times are after the watch starts, and negative times are times before the watch starts.
If a car is at position 0 and is moving in a positive direction, then for times after that (positive times), it will have a positive position. A positive times a positive is positive.
If a car is at position 0 and is moving in a negative direction, then for times after that (positive times), it will have a negative position. A negative times a positive is negative.
If a car is at position 0 and is moving in a positive direction, then for times before that (negative times), it must have had a negative position. A positive times a negative is negative.
If a car is at position 0 and is moving in a negative direction, then for times before that (negative times), it must have had a positive position. A negative times a negative is positive.