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# All Your Base Are Belong to Us

Alignments to Content Standards: F-LE

When Esther tuned into a video game tournament being broadcast online, 64 players remained in the tournament playing the 4-player game. Only one player from each game advances to the next round until a tournament winner is decided. Since Esther tuned in late and missed seeing the beginning of the tournament and details on the total number of rounds, she is trying to figure out how many rounds are left in the tournament.

1. After the round of 64 players (when Esther started watching), how many more rounds will be played in the tournament?

2. Esther creates the function $p(r)=64(b)^r$ to model the number of players p in each round r of the tournament, where r = 0 refers to the round with 64 players. What constant should she use in place of b?

3. Find p(-1). What does this value represent?

4. The announcer mentions that there were initially 4096 players in the tournament. Figure out (by any method you devise) the value of r when p(r) = 4096. What does this value represent?

5. How many rounds are played in the tournament from beginning to end? Explain how you know.

## IM Commentary

The purpose of this task is for students to encounter negative exponents in a natural way in the course of learning about exponential functions.

Part (a) is intended to familiarize students with the context of the problem. Teachers may want to pause the class after students have been given time to work on part (a) to make sure everyone understands the statement of the situation.

If students struggle to determine a value for "b" in part (b), encourage them to organize some related values of r and p in a table. It is likely that some students will get stuck on noticing that "you divide by 4" to find the number of players in the next round. A teacher might suggest that it's tricky to express dividing by the same value repeatedly, so it would be more productive to think in terms of multiplying by some value. "Dividing by 4 is the same as multiplying by...?"

If students are permitted calculators for this task, they may miss out on needing to make sense of evaluating a negative exponent. Without a calculator, they might reason that ${64(}{1\over4})^{-1}=64(4)^1$. Calculator use is a judgment call at the teacher's discretion, but the teacher could still call attention to the result of raising 1/4 to a negative power. The interpretation of negative values for r as referring to rounds that occurred before Esther started watching will still be elicited, even if calculators are used for evaluating.

In part (d), students might count the number of players in rounds backward to 4096. It is not the intention that students use a logarithm in order to solve $64(0.25)^r=4096$, since students may encounter this task well before they know what a logarithm is. Rather, they should use appropriate, familiar tools strategically (MP.5). For example, they could create a table to organize stepping backwards through previous rounds, or they might graph the function p(r) with technology and look at the point (r, 4096). Any representation they create in part (d) will also come in handy for part (e).

## Solution

1. Two more rounds. The next round will have 16 players, and the round after that will have 4 players, from which the tournament winner will be decided.
2. 1/4 or 0.25. 1/4 of the players from any given round move on to the next round. There are different ways to think about it, but one way to reason would be that for r = 1 you want to multiply by 1/4, and for r = 2 you want to multiply by another 1/4.
3. 256. This is the number of players in the round before Esther started watching -- the round immediately preceding the round with 64 players.
4. -3. This means that three rounds took place before Esther started watching.
5. There were a total of 6 rounds of play. Explanations will vary. A student might combine the results from parts (a) and (d)... Three rounds took place before she started watching, the round of 64 is one round, and then two more rounds before a winner was determined, for a total of 6 rounds. Or if a student constructed a table in the course of working through the task, they could simply count the total number of rounds:
 round -3 -2 -1 0 1 2 players 4096 1024 256 64 16 4