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Mathemafish Population


Alignments to Content Standards: F-IF.B.6

Task

You are a marine biologist working for the Environmental Protection Agency (EPA). You are concerned that the rare coral mathemafish population is being threatened by an invasive species known as the fluted dropout shark. The fluted dropout shark is known for decimating whole schools of fish. Using a catch-tag-release method, you collected the following population data over the last year.

# months since
1st measurement
0 1 2 3 4 5 6 7 8 9 10 11 12
Mathemafish
population
480 472 417 318 240 152 103 84 47 32 24 29 46

Task_1_3436be7008e0522a94139d530633c02e

Through intervention, the EPA was able to reduce the dropout population and slow the decimation of the mathemafish population. Your boss asks you to summarize the effects of the EPA’s intervention plan in order to validate funding for your project.

What to include in your summary report:

  • Calculate the average rate of change of the mathemafish population over specific intervals. Indicate how and why you chose the intervals you chose.
  • When was the population decreasing the fastest?
  • During what month did you notice the largest effects of the EPA intervention?
  • Explain the overall effects of the intervention.
  • Remember to justify all your conclusions using supporting evidence.

 

IM Commentary

In this problem, students use given data points to calculate the average rate of change of a function over a specific interval, foreshadowing the idea of a derivative in calculus. Teachers of advanced classes could extend this idea by starting with the average rate of change for the year and then shrinking the interval to approximate the line tangent to the curve at a specific point. Exposing students to these ideas early on could make make it easier for them to grasp calculus concepts when they get to them.

Solution

Solutions are asked to be put in the format of a report. We include below a sample collection of the most relevant pieces of information -- alternatives abound.

Looking at the graph, it seems that there were four intervals in which the rate of population change was significantly different; that is, where the slope of the line appeared to change. These intervals are: (0,1), (1,6), (6,10), and (10,12).

Sol_1_9e7d9d799d0f791be44a6698d4be16d2

\begin{align} m_{(0.1)} &= \frac{\Delta y}{\Delta x} = \frac{472-480}{1-0} = \frac{-8 \text{ fish}}{1 \text{ month}} = -8 \frac{\text{fish}}{\text{month}} \\ m_{(1.6)} &= \frac{\Delta y}{\Delta x} = \frac{103-472}{6-1} = \frac{-369 \text{ fish}}{5 \text{ month}} = -73.8 \frac{\text{fish}}{\text{month}} \\ m_{(6.10)} &= \frac{\Delta y}{\Delta x} = \frac{24-103}{10-6} = \frac{-79 \text{ fish}}{4 \text{ month}} = -19.75 \frac{\text{fish}}{\text{month}} \\ m_{(10.12)} &= \frac{\Delta y}{\Delta x} = \frac{46-24}{12-10} = \frac{22 \text{ fish}}{2 \text{ month}} = +11 \frac{\text{fish}}{\text{month}} \end{align}

The mathemafish population decreased along three of the intervals, but decreased the fastest between 1 and 6 months with a rate of change of –73.8 fish each month. We begin to see the effects of the EPA intervention around the 6th month when the rate of decrease drops from approximately 74 fish per month to 20 fish per month. If not for the intervention, the fluted dropout shark would have decimated the mathemafish population before month 8.

Population in 6th month: 103 fish

Current rate of change: –73.8 fish/month

$$ y = 103 - 73.8x $$

\begin{align} 0 &= 103 \text{ fish} - \left(73.8 \frac{\text{fish}}{\text{month}}\right)\left(x \text{ months}\right) \\ \left(73.8 \frac{\text{fish}}{\text{month}}\right)(x \text{ months}) &= 103 \text{ fish} \\ x &\approx 1.39 \text{ months} \\ \end{align}

$$ 6 + 1.39 = 7.39 \text{ months} $$

Additionally, because of the intervention, by month 10 we even begin to see the population increase with a rate of change of 11 fish/month. If the EPA intervention remains effective, we can expect to see continued growth in the mathemafish population.

Michael Metcalf says:

almost 3 years

In analyzing the solution to this problem, it becomes evident that in essence this was not a mathematically challenging problem. However, the "devil is in the details". How well can one breakdown the interrogatives and directives and give a mathematical interpretation to the problem presented? Better yet, how well and how quickly can a student with a reasonable understanding of the analytical tools of mathematics deconstruct and translate this word problem into some meaningful interpretive solution(s)? Teaching mathematics and training students to respond to and interpret the mathematics within everyday real world scenarios is commendable. I personally have students do this everyday with the aid of an "Inquiry Analytical Algorithm". Believe me practice does not make perfect for 95% of my students. Nevertheless, we persevere and strive to be better problem solvers and critical thinkers. It's all about the learning process. However, the politics of education, "testing and punishing" will make us as educators look foolish and incompetent. I'll be using this problem with the requisite preparatory assistance to challenge my Algebra 2 students. I'll use it as an investigation and discovery problem. They need to understand the challenges before them.