This activity reminds students of previous work they have done with constant speed situations, using for the relationship between distance, rate, and time. This prepares students for representing movement in opposite directions using signed numbers in following activities.
Students may choose to use strategies such as creating a double number line or table of equivalent ratios to make sense of these problems and come up with a solution. While students are free to use these strategies, ensure that they also understand how to use 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 create 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, and follow with a whole-class discussion.
Activity
None
A car is traveling at a constant speed of 60 miles per hour. How far does the car travel in:
2 hours?
5 hours?
hours?
Create a representation that shows the relationship between the elapsed time and the distance traveled for this car.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to remind students of how the equation can be used to solve problems involving movement at a constant speed. To find the distance traveled, we can multiply the rate of travel (or speed) by the elapsed time.
Consider drawing a diagram or 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
Activity
Standards Alignment
Building On
Addressing
7.NS.A.2.a
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
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 negative and a positive number results in a negative number.
Students use their earlier understanding of a chosen zero point, a location relative to this as a positive or negative quantity, and a description of movement left or right along the number line as negative or positive. They extend their understanding to movement with positive and negative velocities and different times. This situation will produce negative or positive end points depending on whether the velocity is negative or positive. Looking at a number of different examples will help students generalize about the sign of the product of a negative number and a positive number (MP8).
This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.
Launch
Use Three Reads to support reading comprehension and sense-making about this problem. Display only the problem stem and the diagram, without revealing the questions.
For the first read, read the problem aloud then ask, “What is this situation about?” (An engineer is recording the speed and direction (east or west) that cars are traveling.) Listen for and clarify any questions about the context.
After the second read, ask students to list any quantities that can be counted or measured. (the number of cars that pass by, the average speed of each car, the number of cars traveling east or west, the number or cars traveling faster or slower than the speed limit)
After the third read, reveal the first question: "A car is traveling east at 12 meters per second. Where will it be 10 seconds after it passes the camera?" and ask, “What are some ways we might get started on this?” Invite students to name some possible starting points, referencing quantities from the second read. (We can draw a diagram on the number line. We can calculate how far the car will have traveled in 10 seconds.)
Give students 6–7 minutes of quiet work time, and follow with a whole-class discussion.
Representation: Internalize Comprehension. Begin with a physical demonstration of people moving with a constant speed in two different directions to support connections between new situations and prior understandings. Consider using the prompts: “Does this situation remind anyone of something we have done before?” “What factors determine the person’s final position?” Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
None
A traffic safety engineer was studying traffic patterns. She set up a camera to record the speed and direction of cars and trucks that passed by. She decided to represent positions to the east of the camera with positive numbers and positions to the west of the camera with negative numbers.
Number line. To the left is the word west. To the right is the word east. The numbers negative 50 through 50, in increments of 10, are indicated. There are tick marks midway between.
A car is traveling east at 12 meters per second. Where will it be 10 seconds after it passes the camera?
A car is traveling west at -14 meters per second. Where will it be 10 seconds after it passes the camera?
Complete the table to show the position of each vehicle after traveling at a constant velocity for the given amount of time.
velocity
(meters per
second)
time after passing
the camera
(seconds)
position
(meters)
equation
car A
+12
+10
+120
car B
-14
+10
car C
+9
+5
car D
-11
+8
car E
-15
+20
car F
+8
0
Complete the sentences. Be prepared to explain your reasoning.
A positive number times a positive number equals a _______________________.
A negative number times a positive number equals a _______________________.
Student Response
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Building on Student Thinking
Encourage students who get stuck to use the provided number line to represent each situation.
Activity Synthesis
The purpose of this discussion is to emphasize that the product of two positive numbers is a positive number and that the product of a positive number and a negative number is a negative number. Begin by displaying the table and the number line from the Task Statement for all to see. Invite students to share their responses and reasoning for each car.
Demonstrate how cars with a positive velocity are moving towards the east and cars with a negative velocity are moving towards the west. Then place a point on the number line to represent each car’s position (except Car E’s position) at the given time. Discuss the following questions:
“Where would Car E be located?” (At the position -300 to the east of the camera.)
“What do you notice about the product of two positive numbers?” (The product is also positive.)
“What do you notice about the product of a positive number and a negative number?” (The product is negative.)
“What does it mean for Car F to be at the position 0 in this situation?" (Car F is traveling east at a velocity of 8 meters per second. Its position is measured 0 seconds after passing the camera, meaning that its position is measured exactly as it is passing the camera.)
14.3
Activity
Standards Alignment
Building On
Addressing
7.NS.A.2.a
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
In this activity, students consider a context of vehicles traveling past a traffic camera at varied constant velocities and in different directions. Students reason about what it means to have a negative time (the vehicle has not yet passed the traffic camera) or a negative velocity (the vehicle is traveling from east to west) in this context. They engage in reasoning abstractly and quantitatively as they represent velocity, distance, and time relationships using an equation, then interpret the results in terms of the situation (MP2).
Students also use repeated reasoning to describe a pattern for identifying the sign of the product of two negative numbers (MP8).
The Launch mentions that this activity is the same context as one in the previous lesson. In this course, the same context of the traffic camera is used in the previous activity of the current lesson.
During the Launch, display this image for all to see:
Ask, “Where was the person 5 seconds after this picture was taken?” (somewhere to the right of the person’s current position) and “Where was the person 5 seconds before this picture was taken?” (somewhere to the left of the person’s current position).
If necessary, point out that if we assume the person is walking at a constant speed, their before and after locations should be equally far from their current position in the image. Ask students how they might represent time in this situation. (5 seconds before the picture was taken could be -5, while 5 seconds after the picture was taken would be +5. The picture was taken when the time was 0.)
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 whole-class discussion before continuing to the next question.
MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context. Advances: Reading, Representing
Representation: Access for Perception. Begin by showing a video of a car or person moving forwards, and then play the same footage in reverse to support understanding of the context. Supports accessibility for: Conceptual Processing, Language
Activity
None
A traffic safety engineer was studying traffic patterns. She set up a camera to record the speed and direction of cars and trucks that passed by. She decided to represent positions to the east of the camera with positive numbers and positions to the west of the camera with negative numbers.
Number line. To the left is the word west. To the right is the word east. The numbers negative 50 through 50, in increments of 10, are indicated. There are tick marks midway between.
A car was traveling east at 12 meters per second. Where was the car 10 seconds before it passed the camera?
A car was traveling west at -14 meters per second. Where was the car 10 seconds before it passed the camera?
Complete the table to show the position of each vehicle after traveling at a constant velocity for the given amount of time.
velocity
(meters per
second)
time after passing
the camera
(seconds)
position
(meters)
equation
car A
+12
-10
-120
car B
-14
-10
car C
+9
-6
car D
-11
-9
car E
-15
-4
car F
+8
-13
Complete the sentences. Be prepared to explain your reasoning.
A positive number times a negative number equals a _______________________.
A negative number times a negative number equals a _______________________.
Activity Synthesis
The key thing for students to understand from this discussion is that a negative number multiplied by another negative number results in a positive number. Begin by displaying the number line and table from the Task Statement for all to see. Invite students to share their responses, and record them in the table.
Then select a car whose equation has a positive and a negative factor, and demonstrate its path on the number line. For example, explain that Car A is traveling from west to east since its velocity is positive. Since its time is represented by a negative number, the car has not yet passed the camera and, therefore, must be located on the left side of the camera, making its position a negative number.
Then select a car whose equation has two negative factors. For example, Car D has a negative velocity, which means that it is traveling from east to west. It also has a negative time, which means it has not yet passed the camera. Car D must be located to the right of the camera, making its position a positive number.
Ask students to complete the sentences in the last problem. Consider posting these in the classroom for all to see for the next few lessons.
Lesson Synthesis
To help students consolidate their learning about multiplying signed numbers, consider asking:
“How can we represent how fast something is moving to the left or right from a starting point?” (We can pick one direction to represent with a positive velocity, and then the other direction has a negative velocity.)
“If an object moves at a velocity of 4 meters per second for 5 seconds, what is its finish point? Why?” (20 meters in the positive direction from the starting point, because )
“If an object moves at a velocity of -4 meters per second for 5 seconds, what is its finish point? Why?” (20 meters in the negative direction from the starting point, because )
“What does it mean if a time is negative?” (It means the amount of time before a chosen reference time.)
“What kind of number do we get when we multiply a positive number by a negative number? Use a situation from the lesson to explain why this makes sense.” (We get a negative number. For example, if you were walking from left to right, then before you crossed the reference point, you would have been to the left.)
“What kind of number do we get when we multiply a negative number by a negative number? Use a situation from the lesson to explain why this makes sense.” (We get a positive number. For example, if you were walking from right to left, then before you crossed the reference point, you would have been to the right.)
Student Lesson Summary
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.
A number line with the numbers negative 10 through 10 indicated. A point is indicated at zero and is labeled "reference point." Another point is indicated at negative 4 and is labeled "4 units to the left of zero." A third point is indicated at 7 and is labeled "7 units to the right of zero."
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 50 meters to the right of zero. In other words,
If you start at zero and move -5 meters per second for 10 seconds, you will be 50 meters to the left of zero. In other words,
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.
Three points are labeled on a number line. The numbers negative 10 through 10, in increments of 1, are indicated. A dot at negative 4 is labeled "4 seconds before the start time". A dot at 0 is labeled "start time". A dot at 7 is labeled "7 seconds after the start time".
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.
Number line. 7 evenly spaced tick marks. Scale negative 15 to 15, by 5's. Three equal sized arrows pointing to the right, one from 0 to 5, 5 to 10, and 10 to 15.
For times before that (negative times), it must have had a negative position.
A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing right and a dot are plotted. The first arrow points from negative 15 to negative 10. The second arrow points from negative 10 to negative 5. The third arrow points from negative 5 to 0. A dot is above 0.
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 blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 0 to negative 5. The second arrow points from negative 5 to negative 10. The third arrow points from negative 10 to negative 15. A dot is above 0.
For times before that (negative times), it must have had a positive position.
A blank horizontal number line from negative 15 to 15 by 5’s. Above the number line with three arrows pointing left and a dot are plotted. The first arrow points from 15 to 10. The second arrow points from 10 to 5. The third arrow points from 5 to 0. A dot is above 0.
Standards Alignment
Building On
Addressing
7.RP.A
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.
