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).

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

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.)  

The purpose of this discussion is to reinforce these key ideas:

• The product of two numbers with different signs is negative.
• The sum of two numbers with different signs will be the same sign as the number with the largest magnitude (or the sum will be 0 if the two numbers have the same magnitude).

Invite students to share their responses and reasoning to the fourth question asking how many parks and clam beds a city would need to absorb the carbon dioxide released by 400,000 cars. As students share their combinations of parks and clam beds, consider discussing the following questions:

• “How did you calculate the amount of carbon dioxide released by the parks?” (I multiplied the average amount of carbon dioxide released by one park and multiplied it by the total number of parks.)
• “Why is the carbon dioxide released by the parks a negative number? How does that make sense in this situation?” (A positive number multiplied by a negative number results in a negative number. This makes sense because trees absorb carbon dioxide.)
• “How do we know when the number of parks and clam beds will be enough to absorb the carbon dioxide released by the cars?” (The sum of all the numbers will be 0.)
• “How can a set of numbers have a sum of 0?” (The total magnitude of the positive numbers is the same as the total magnitude of the negative numbers.)