Rate Paradoxes

The focus of this Pull-Out is on computing rates and average rates. Calculations involving rates can often be counterintuitive.

In Activity 1, the rate used is speed (distance/time). Students discover that the relationship between speed and time when the distance is held fixed is not linear. They also learn that the arithmetic average of two speeds does not always give the correct average speed for a trip.
In Activity 2, the rate is fuel efficiency (miles/gallon). Here students discover that it's possible for a smaller increase in fuel efficiency to result in a larger savings in gasoline.
In Activity 3, the rate is batting averages. Students are challenged to find a situation in which one player can have lower batting averages the first and second half of a season compared to another player, and yet have a better batting average for the season.

Mathematics prerequisites and discussion:
Students should be familiar with the concept of a rate. The rates used in this Pull-Out are speed (miles/hour), fuel efficiency (miles/gallon), and batting averages (Number of hits/Number of times at bat). They should be able to add fractions or use a calculator to do so. They should know how to calculate an arithmetic average (for example, the arithmetic average of 3, 6, and 10 is (2 + 6 + 10)/3 = 6).

Materials needed:
Calculators.