8.1: Distance, Rate, Time (5 minutes)
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.
- 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?
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.
8.2: Going Left, Going Right (10 minutes)
The purpose of this activity is to understand that a rate of travel at a constant speed (defined as velocity) can indicate the direction of travel, by using a negative or positive value to describe travel to the left or to the right of a location taken to be 0.
Students use their earlier understanding of a chosen zero point and describe their movement left (negative) or right (positive) along the number line, with different speeds. This will produce negative or positive end points depending on if they are moving to the left or the right. This will lead to students describing negative numbers multiplied by positive in the next activity.
Remind students we have seen in earlier lessons that we can pick a location to represent zero, and then locations on either side are positive or negative.
Supports accessibility for: Visual-spatial processing
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
- After each move, record your location in the table. Then write an expression to represent the ending position that uses the starting position, the speed, and the time. The first row is done for you.
expression 0 right 5 3 +15 \(0 + 5 \boldcdot 3\) 0 left 4 6 0 right 2 8 0 right 6 2 0 left 1.1 5
- How can you see the direction of movement in the expression?
- Using a starting position \(p\), a speed \(s\), and a time \(t\), write two expressions for an ending position. One expression should show the result of moving right, and one expression should show the result of moving left.
Students may want to use the number line to help them with the position changes in the table.
Have students write in words how they calculated ending positions for the left and right if they get stuck trying to write an expression with variables.
Ask students to share how they could see the direction of travel in their expressions. It has to do with whether they added or subtracted the product \(st\) from zero. Ensure that everyone understands why \(p+st\) represents final positions to the right of zero and \(p-st\) represents positions to the left of zero.
We saw, earlier in this unit, that subtracting a positive is the same as adding a negative. So let’s write a single expression, \(p+vt\) where instead of speed (which is always a positive number) we use a signed number for speed plus direction and we call this quantity velocity. Velocities when moving to the right will be represented by positive numbers, and velocities when moving to the left will be represented by negative numbers.
8.3: Velocity (15 minutes)
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.
Supports accessibility for: Conceptual processing; Visual-spatial processing
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.
time after passing
+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?
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.
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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.
Design Principle(s): Support sense-making; Optimize output (for generalization)
- 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.
- A negative number multiplied by a positive gets a negative product.
- 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?
- What kind of number do you get when you multiply a negative number by a positive number?
8.4: Cool-down - Multiplication Expressions (5 minutes)
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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.
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,
\(\displaystyle \text-5\boldcdot 10 = \text-50\)
In general, a negative number times a positive number is a negative number.