6.3.3 Converting & Estimating Measurements

Ms. Rykken's students were excited about the recent CO₂ car competition in their industrial technology class, where they raced the cars they had designed and built. She decides to use this opportunity for them to use their own data and to review working with rates and unit conversion.

She began the class by asking for the velocity of the winning car. Students could only respond with the winning time (0.971 seconds).

Teacher: How is the velocity of real cars expressed?

Student: Speedometers tell us how many miles per hour we're going.

Teacher: That sounds like a relationship between distance and time. What is the relationship between distance and time for the winning CO₂ car?

Student: Well, the track was 66 feet long and it took the car 0.971 seconds to travel that distance.

Teacher: So the winning car traveled 66 feet in 0.971 seconds. We can write that relationship as a ratio, or fraction $\frac{66\ feet}{0.971\ seconds}$ . But you said that real cars express velocity as miles per hour. How can we convert feet per second to miles per hour?

Student: The feet need to be changed to miles, and the seconds need to be changed to hours.

Teacher: Yes, but how do we do that?

Student: I know how to change the time. There are 60 seconds in every minute, and 60 minutes in every hour. We can just multiply.

Teacher: Tell me more.

Student: If I multiply $\frac{66\ feet}{0.971\ seconds}\times \frac{60\ seconds}{1\ minute}\times \frac{60\ minutes}{1\ hour}$, I'll know how many feet the car went in 1 hour.

Teacher: How do we know that multiplying by those fractions didn't change the original velocity?

Student: Because each of the fractions is equivalent to 1, and multiplying by 1 gives you the same value.

Teacher: Since the fractions are equivalent to 1, could I have inverted them and gotten the same result from multiplying?

Student: No, that would mess up the units. The fractions need to be set up so that the units cancel, like seconds with seconds and minutes with minutes. After multiplying, you're left with $\frac{237,600\ feet}{0.971\ hour}$.

Teacher: That makes sense. Now we've converted the time, but we're not finished. For real cars, we express velocity in miles per hour. How do we convert the feet to miles?

Student: The same way. We multiply using the relationship that every mile has 5,280 feet.

Teacher: Do I multiply by $\frac{5,280\ feet}{1\ mile}$ or $\frac{1\ mile}{5,280\ feet}$?

Student: If you want the feet to cancel, you'll need to use $\frac{1\ mile}{5,280\ feet}$ .

Teacher: Like this? $\frac{237,600\ feet}{0.971\ hour}\times \frac{1\ mile}{5,280\ feet}$?

Student: Yes.

Teacher: What is the result?

Student: About 46.3 miles per hour.

Teacher: WOW! That's amazing! It's a good think you didn't race your cars outside the school, because the speed limit there is 30 miles per hour.

Student: Could we have changed the feet to miles first and gotten the same answer?

Teacher: Good question. Does it change the result if we multiply in a different order?

Student: No, multiplication is commutative, which means that the order doesn't matter.

Teacher: Yes, that's right.

Student: I did all those steps at the same time and got the same result.

Teacher: Like this:  $\frac{66\ feet}{0.971\ seconds}\times \frac{60\ seconds}{1\ minute}\times \frac{60\ minutes}{1\ hour}\times\frac{1\ mile}{5,280\ feet} $? Excellent! That's very efficient. Now I'd like each of you to find the velocity of your CO₂ car in miles per hour.

Ms. Rykken circulates the room to assist students and hear tales of their race experiences.