Baseball Cards

Task

A student has had a collection of baseball cards for several years. Suppose that $B$, the number of cards in the collection, can be described as a function of $t$, which is time in years since the collection was started. Explain what each of the following equations would tell us about the number of cards in the collection over time.

1. $B=200+100t$
2. $B=100+200t$
3. $B = 2000-100t$
4. $B=100-200t$

IM Commentary

The statement of the question is appropriate for an instructional setting but not specific enough for an assessment item. In fact, this task could be put to good use in an instructional sequence designed to develop knowledge related to students’ understanding of linear functions in contexts. Though students could work independently on the task, collaboration with peers is more likely to result in the exploration of a range of interpretations. In a follow-up class discussion, the teacher could help students explore the limits of possible interpretations and speak to setting realistic parameters for mathematical models.

Solution

1. In this equation, we can first observe what $B$ looked like at the start of the collection, which is $t = 0$ years. This gives

   $$ \begin{align} B &= 200 + 100(0) \\ B &= 200 \end{align} $$

   We see that the student started out with 200 baseball cards. As $t$ increases, the number of baseball cards increases at a rate of 100 cards per year, meaning the student added 100 cards to his collection each year.

2. At the start of the collection, $t = 0$, we have

   $$ \begin{align} B &= 100 + 200(0) \\ B &= 100 \end{align} $$

   We see that the student started with 100 baseball cards. The number of baseball cards increases at a rate of 200 cards per year, meaning the student added 200 cards to his collection each year.

3. We observe that at the start of the collection, when $t = 0$, we have

   $$ \begin{align} B &= 2000 - 100(0) \\ B &= 2000 \end{align} $$

   So at the start of the collection, the student had 2000 cards. However, in this case, the size of the collection decreases with time. As $t$ increases, $B$ decreases at a rate of 100 cards per year; that is, the student loses 100 cards each year. As the amount of time that has passed is only "several years," this equation may be valid. However, after 20 years, $B$ will become negative, in which we would either say that the equation no longer applies or possibly interpret it as meaning that he owes someone baseball cards (an unlikely scenario but it is consistent with interpretations in other contexts where negative numbers represent debt).

4. At $t = 0$, the start of the collection, we have

   $$ \begin{align} B &= 100 - 200(0) \\ B &= 100 \end{align} $$

   The student started his collection with 100 cards. However, after just one year, $t = 1$, we have

   $$ \begin{align} B &= 100 - 200(1) \\ B &= -100 \end{align} $$

   This says that after one year, the student has -100 baseball cards. As in part (c), this could conceivably mean that he owes baseball cards to someone else, although this seems like a highly improbable situation.