Explore the National Parks through Measurement MEGAN H. WICKSTROM ELIZABETH W. FULTON

How would you solve this task? Suppose you have a map of Montana. When you look at the map, you see that the scale is 1 inch: 100 miles. When you trace the outline of Montana onto ¼ inch graph paper (the grid lines are spaced ¼ inch apart), you count that Montana takes up about 235 squares. Using this information, approximately, what is the area of the state of Montana?

Background of the Task  Given in a k-8 Geometry and Measurement Course at MSU: Aligns with 4 th grade CCSSM Measurement standards.  The course is activity based. Students solve, present strategies, and discuss misconceptions.  This problem elicits many misconceptions from students…  What are they struggling with?  How can I make it meaningful?  How many students initially solved this correctly?

Background of the Task  Only 4 out of 50 pre-service teachers solved this task correctly.

What were some common misconceptions? Applying Length Reasoning to Area  Students assumed that if 1 in = 4 quarter inches, then 1 square inch = 4 quarter inch squares  Students assumed that if 1 inch represents 100 miles that 1 square inch must also represent 100 square miles

What were some common misconceptions? Applying Length Reasoning to Area

What about if they made a drawing?  Number of squares was off  Applies length reasoning to Drawing  Correct drawing but cannot articulate what it means in terms of area

Secondary Misconception: Should I multiply or divide? ¼ inch is 235 squares. To find what 1 inch would be, we multiply 236 by 4 and see that 1 inch would be 940 squares.

Correct Responses: Estimation Montana has approximately an area of 150,000 square miles. 4 quarter inches make up 1 inch and 1 inch is 100 miles. I counted 235 tiles and found 2 numbers to divide 235 by in order to get the length of one side of MT. I counted 20 squares across, which turned out to be 5 inches of 500 miles. Then I counted squares down which is close to 300 miles. Now in order to find the area, I multiplied 500 x 300.

Correct Answer: Finding Area of ¼ inch square

New Task Your task today is to find the area of Yellowstone National Park using the map you are given and any of the following tools (please be ready to present your solution at the end of class):  Modeling Dough  Graph Paper (1/8, 1/4, ½, and 1 inch sizes)  1-inch tiles  String  Weighing Scale Describe how you found the area of the park and the tool you used. What did you do first, second, third, etc? and why?

Imagine how you might solve the task using the tools. What could you discuss with students?  Modeling Dough  Graph Paper (1/8, 1/4, ½, and 1 inch sizes)  1-inch tiles  String  Tracing Paper  Weighing Scale

Student Strategies: Modeling Dough  Students thought about making the park from dough and moving the pieces around to make a rectangle.  The playdough was hard to use and many groups abandoned this strategy.  Discussed keeping the same thickness

Student Strategies: Graph Paper/Tracing Paper  Students discussed size of unit and how the unit relates to the scale.  Smaller the unit, the more precise we could be  Either traced or glued units down

Student Strategies: Tiles Many students found it very simple and straight forward to use the tiles. They had to think about what 1 tile represented and how they might compensate for the small spaces not covered.

Student Strategies: String  String was thrown in as a trick  Students used string to approximate the length and width of the park and form other shapes.  We discussed how shapes with the same perimeter do not necessarily have the same area.

Teacher Strategy: The Scale  No one ever tries to use the scale, so I decided to try.  I weighed the map against a 1-inch tile to see if I could create a proportion to compare the area.  The scale was not precise enough.

Getting at Comparing Units with Extension Tasks  Yellowstone Lake is the largest freshwater lake above 7,000 ft in North America. In your packet, you also have a map of Lake Michigan. Your task is to describe and compare the surface areas of both lakes.  When people think of National Parks, they often think of Yellowstone National Park and Yosemite National Park. Yosemite had about 4 million visitors last year while Yellowstone had about 3 million visitors. Your task is to compare the two parks.  Volcanos are kind of scary. In 1980 Mt. St. Helens erupted and spewed ash all the way to Bozeman. Did you know we live pretty close to a super volcano? Your task is to compare the caldera in Yellowstone to Mt. St. Helen’s caldera.

More Extension Tasks  Sometimes it is hard to imagine how big our state, Montana is. Your task is to compare Yellowstone National Park to the state of Montana. How big is one compared to the other.  It is very hard to imagine how big our country is compared to Yellowstone National Park. Compare Yellowstone to the map of the United States. What proportion of the country does Yellowstone take up?  You have a very different type of task then the other groups. Yellowstone is a very accessible National Park in that you can drive a lot of it. If I never wanted to leave my car, estimate what proportion of the park I might be able to see (using the topographical map might help).

Takeaways from the Task  Students often struggle with finding and comparing lengths and areas without physical tools. The tools allowed the students to make sense of the task in an appropriate way.  It allowed the instructor to discuss area, area units, comparing areas, and how units are related to each other.  Students developed ideas of reasonableness in terms of area.  Students developed multiple strategies to use and compare areas.

Questions?? Thank you!