Question: How Do Animals Travel Over the Surface of the Snow?

Snow Travel

The hook:  First: A snowshoeing trip with students gets them thinking about why snowshoes are effective tools for humans to use when snow travelling. (A picture might serve the same purpose.)  Second: The students observe the tracks of various animals that are “floating” over the snow. They also watch, while eating their lunch, a large moose travelling past the hut sinking deeply into the snow. (Again, a photo or video clip might serve the same purpose.)

Question: How do animals travel over the surface of the snow?

Class hypothesis: Foot size and body weight affect the ability of animals to travel over the surface of the snow.

How can we prove mathematically that foot size and body weight affect the ability of animals to travel over the surface of the snow?

Determine more questions that need answering: 1. What animals are good snow travelers? Discovery: Some effective snow travelers are snowshoe hare, lynx, wolverine, mice and squirrels. Poor snow travelers are moose, deer, elk and humans (without snowshoes). 2. What about: Wolves? Coyotes? Cougars? Raccoons? Birds? Other?

Refined class hypothesis : If an animal has a large foot and a small body weight, they will travel well over the surface of the snow.

Sub-question: How can we figure out the foot size of a particular animal?

 Student idea: Take a string and trace around the outside of the track. Whichever string is longest is the biggest track. Discussion…  Student idea: Draw the track on grid paper and count the squares. Discussion…  Student idea: Make a box around the track on the grid paper. Discussion…  Student idea: Then you could just do length X width of the track. Discussion…  Teacher question: Is using length X width an accurate way to measure all tracks? What about animals with finger- like extensions? (birds, raccoons) Will they be better or worse snow travelers? Discussion…

Problem 1:Given the following materials (string, graph paper, ruler, scissors) determine the area of one of the different tracks selected from the box. Animal:…………………………………………………………………………….

Describe how you determined the area of the irregular shape.

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Problem 2: If both track area and weight are important, how would you rank the animals in each of the following sets? (1 = Best Floater)

Group 1 Animal Weight (kg) Track Area (cm2) Rank A 15 5 B 20 5 C 25 5 D 15 10 E 20 10 F 25 10

Group 2 Animal Weight (kg) Track Area (cm2) Rank A 10 5 B 15 5 C 20 5 D 25 5 E 10 10 F 15 10 G 20 10 H 25 10 Which are easiest to compare? Hardest? Problem 3: How can we compare Animals A & E in terms of sink-ability? E has more mass, but also has more foot area to make up for it.

Animal Weight (kg) Track Area(cm2) Rank A 15 5 B 20 5 C 25 5 D 15 10 E 20 10 F 25 10

3 2 2 2

3 3 2 2 2

Describe how we could consider the impact of both track area and weight in relation to “sink-ability” using the example from above. What do we need to keep in mind?

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Assumptions: Describe the assumptions that we are making in this model of an animals foot?

…………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… ………… Analyzing “Sink-ability” Data (Note: If you do not know how to compare decimal numbers and this is something that distracts you from the main mathematical emphasis of this lesson, you can use the spreadsheet’s sort function to put the ratios in order. The sort allows more fluent access to information that is relevant to this investigation.)

Task 1: Complete the table by calculating the weight (w) to track area (T) ratio (w/T) for all the animals

Task 2: Predict if the animals will sink (0) or float (1).

Describe how you predicted if an animal would sink or float.

…………………………………………………………………………………………… …………………………………………………………………………………………… …… Describe the assumptions that we are making in this model of an animals foot?

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What ratio marks the division between floaters and sinkers? Describe how you made that decision.

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Describe other factors that might contribute to floating or sinking.

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Describe how you might adjust this model to make it a more accurate representation.

…………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… …………………………………………………………………………………………… ………… Using Graphical Representations for “Sink-ability” Data

Problem: How does an x-y scatterplot combine 2 values? Does it allow easy comparison of all data points?

If our hypothesis is correct, what areas of the graph should have the best snow travelers? The worst?

Does the actual animal data match our prediction?

Without calculating, can we use the position of the dots on the graph below to rank some of the animals? All of them? Use the data table to check your ideas. (Note: You can manipulate the graph and data in the spreadsheet file.)

Where on each graph would you expect to find high (H) , medium (M) , and low (L) sinkage?

We 4 igh t 3 (kg ) 2

1 2 3 4

1 2 3 4 5

Track Area (cm2) Teacher Notes: Ranking “Sink Factors”

- This is a mathematical inquiry. We already know how the animals rank, so ranking them isn’t really the point of the question. The point is to understand how ranking shows up on a scatterplot. In this sense, any set of ratios could be used; e.g. mass and volume as they combine to form density; wheel and axle diameter as they combine to impact the speed of a mousetrap-powered car, etc.)

- The question isn’t one that is typically asked of scatterplots. It forces kids to think outside of familiar procedures that might allow them to bypass deeper consideration of the ideas.

- Our knowledge of how the animals rank serves our investigation into the scatterplot; the context serves the math. We can use it to check our ideas re: how the data points allow ranking.

- The investigation provides an opportunity for students to make and test their own ideas about how a scatterplot works.

- Feedback is immediate, and there’s no need for outside authority.

- Students can invent hypothetical data sets that allow easier comparison; e.g. How do animals with the same ratio appear in relation to one another on the graph?

- Spreadsheet technology makes it easier for (a) students to make and test ideas about hypothetical data and (b) manipulate the data so that it is easier to interpret. In other words, the technology can be used to enhance mathematical inquiry, not just to report results in a different way or to do something in a virtual way that could just as easily be done with manipulatives or on paper.

- The investigation doesn’t result in knowledge of a new procedure or concept as typically defined. But it can deepen and broaden understanding of ratio in a way that allows easier connection to other situations where it might be relevant (e.g. trig).

- The investigation starts from a place of puzzlement about a big idea. When students grapple with how to compare “A and E,” they realize that both track area and weight are significant and that it’s hard to rank “sink factors” when both change: Does the area go up by enough to compensate for the extra weight? This forces them into the space of perplexity that typical procedures for comparing ratios resolve (e.g. making x-y scatterplots, converting to equivalent ratios, converting to rates such as g/cm3, etc.). By deeply understanding the question or the problem, they already have a much better understanding of the answer.