By Brynja Kohler and Joyce Smart

Muskoxen and Wolves

By Brynja Kohler and Joyce Smart

Description

This is an integrated science/math lesson that leads students to interpret linear models about arctic animals. The given models are presumably based on based on real scientific data, and students will have to describe the process that could have led to the model result. Students will practice various skills associated with graphing and identifying slopes and intercepts, and they will also learn about behaviors of muskoxen and wolves.

Standard

F.LE.5: Interpret the parameters in a linear or exponential function in terms of a context.

Objective

Students interpret slope and intercepts of a regression line within a scientific context, describe implications and limitations of the linear model, and can explain how data collection led to the model formulated. (comprehension-and-communication)

Launch

Introduce this task by asking students what they know about wildlife in the arctic tundra. Ask them to make notes of their observations while you play the following video-

Muskoxen and wolves video link (Note: Use download-helper (with Firefox) to download the video)

Call on students to share their observations. Some responses might include noticing that

- the wolves are white and very obvious in the summer,

- both the wolves and the muskoxen travel together in groups,

- there isn’t much to eat except tundra grass for the muskoxen (they are vegetarian),

- the muskoxen surround their young to protect them against the wolves,

- the large group of muskoxen appeared to be confused by the wolves, while a small group seemed to stand their ground fairly well,

Explore

Have students work in pairs on the muskoxen tasksheet (page 2). Pass out graph paper and rulers (or use calculator graphing functions) so students can practice making appropriately scaled graphs. It is okay for students to make really rough sketches in response to prompt “a”, but they should be able to explain to one another that for both summer and winter models, scientists must have counted wolves and recorded the size of the observed muskox group. The data points would be each observation (wolf sightings (per x hours), muskoxen group size) and a trend line was computed to ‘best fit’ the data by minimizing the sum of squared deviations.

Discuss

Invite select pairs of students to share their work. Bring out that the rate of wolf sightings per hour is a different quantity than the slope of the line – how can the slope of the line be interpreted as a rate? Compare results of the student work with the key provided and discuss scientific interpretations of the model.

Muskoxen and Wolves

The size of a muskoxen group increases linearly with increasing wolf density. Muskox group size also is different in summer and winter. After collecting much data the statistical line of best fit (the regression line) for the data was computed to be

y =6.3x + 16.76in winter and

y = 5.46x + 7.76in summer.

Here y is the muskox group size and x is the number of wolf sightings per hour. (Note: The actual data was collected by counting the number of wolf sightings in 100 hours, then averaged to give the per hour estimate.)

1. Make a sketch of a scatter plot of plausible data scientists may have collected before coming up with the regression line. Explain their procedure for arriving at the regression line.
2. For each equation, identify the slope. What does the slope mean? What are its units?
3. If there are no wolf sightings what is the muskox group size? Does the model make sense?
4. What happens to the number of muskox in a group when the wolf sightings per hour increase? Why would this happen?

Now, plot these two lines and estimate:

1. the number of muskox in a group with 3 wolf sightings per hour in the summer,
2. the number of wolf sightings estimated if the muskox group is 30 in the winter.

1. How much larger is the muskox group size if 200 wolves are sighted per 100 hours in the winter as compared to 200 wolves sighted per 100 hours in the summer?
2. What assumptions does this model make? What are some limits on the interpretation of the model?
3. What information do you gain from examining graphs, from tables?

KEY for the Muskoxen and Wolves Tasksheet

1. An example of a plausible plot

1. For each equation, identify the slope.

6.3 winter, 5.63 summer

What does the slope mean?

muskox group size increases by 6.3 for each additional wolf-sighting per hour in winter

muskox group size increases by 5.63 for each additional wolf-sighting per hour in summer

What are its units?

muskoxen per wolf-sighting per hour

1. If there are no wolf sightings what is the muskox group size? Does the model make sense?

16.7 in winter and 7.6 in summer

The model makes sense given that these are averages, but you wouldn’t see 16.7 muskox in any group ever. The value is positive though, so it is not out of the range of a reasonable interpretation of data.

1. What happens to the number of muskox in a group when the wolf sightings per hour increase? Why would this happen?

The muskox in a group also increase. This makes sense since they would gather together to defend themselves and/or protect the young.

Now, plot these two lines and estimate:

1. the number of muskox in a group with 3 wolf sightings per hour in the summer,

24ish muskoxen

1. the number of wolf sightings estimated if the muskox group is 30 in the winter.

2.1 per hour, 21 per 10 hours, 210 per 100 hours.

1. How much larger is the muskox group size if 200 wolves are sighted per 100 hours in the winter as compared to 200 wolves sighted per 100 hours in the summer?

29.36 compared to 18.68. (29.36-18.68)/18.68 = .57 so 57% larger in winter compared to the summer.

1. What assumptions does this model make? What are some limits on the interpretation of the model?

Linearity of the data, averaging of data - so the values in the model are smoothed out where in the real world there would be considerable variability, there may be a maximum number of muskox in any area, and there may be some limit to the number of wolves per hour.

The models do not provide information about how the muskox group size varies in the time frame between winter and summer or at different times in the winter and summer. They also do not show how the muskox population varies with the total population of wolves in a region. It is not clear how space is accounted for. Do scientists follow a muskox group? We can infer from the models that there is a linear correlation, but we don’t know about causation. Which is variable is dependent and which is independent: do the muskox get together because of the numerous wolves in the area, or do the large groups of muskoxen attract more wolves?

1. What information do you gain from examining graphs, from tables?

The graphs reveal that there isn’t an intersection with real world meaning – that is, there is not a particular amount of wolf-sightings that lead to the same number of muskoxen in winter and summer. The graph shows the winter group is always more than the summer group.

Reasoning from a table is helpful to create the graphs and make estimates like the questions above.

The problem in this lesson comes from Biomath: Problem Solving for Biology Students, by Robert W. Keck and Richard R. Patterson. Copyright 2000, by Addison Wesley Longman, Inc., San Francisco, CA.

Image retrieved from on September 29, 2011.

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