Intermath: Investigations: Algebra: Patterns
A professional bass fisherman caught 385 bass during a 14-day tournament. Each day he caught three more fish than he did the day before. How many fish did the fisherman catch on each individual day?
In beginning to answer this problem, it is important to consider underlying questions that may need to first be answered before attacking the original question of how many fish were caught each day. There are two things that are unknown in this problem: 1) how many fish were caught on the first day, 2) how many fish were caught each day after. It is important to determine how many fish were caught the first day, because it is obviously not zero.
One way to solve this equation is simply by charting the values beginning at the end. Knowing that on the 14th day 385 fish were caught, to determine the number caught on the 13th day it is important to remember three less fish will be caught than on the 14th day. And, here begins our pattern:
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So from working from the end to the beginning, it is clear that on day one of the trip he caught 346 fish. And our table, if looked at from the bottom up, shows that each day there after he caught three more fish than he had the day prior. The pattern of adding three for each additional day to the previous total of fish caught is revealed clearly in this chart. Because there is a distinct pattern, maybe a formula could be used to shorten the development of the pattern.
To determine how many fish were first caught, the following equation can be utilized:
in this equation the unknown is x, which is equal to the number of fish caught the on the nth day. The n represents on which day the fisherman is fishing in reference to the first day. For instance, the last day would be day 14 and the first day would be day 1. The number 3 is multiplied by n to determine the number of additional fish caught according to the n number of days. One is subtracted from n because on the first day three additional fish are not caught. Day one is simply the orginal number of fish caught, the +3 compounds from there for the next 13 days. Consequently, when n=14, meaning the last day, the equation will produce (x + 3(14-14)=385) = (x + 3(0)=385) and it is clear that on the last day the number of fish cauht x = 385, which is defined in the equation. By substituting various reasonable n's, the information in the graph can be rediscovered.
This pattern and equation can be explored in many different applications. For instance, Excel.
*Be sure to note in this application the formula used to calculate the total number of fish caught each day. You will find that it is the absolute value of 3(14-A2)-385. In this rewrite of the equation, x became negative as it was moved to the other side of the equal sign. To retain the identity of the equation (ie. not changing every sign), I decided to take the absolute value of the equation, eliminating the negative from the answer.