Ethan went on a kayaking expedition down the Ottauquechee River last week. He left school at 2:35 and paddled downstream 3 miles until he hit the sewage treatment plant at 3:05. He decided to get out of the river at that point (wise decision) and went across the street to Pizza Chef. He then headed back up the river in his boat. It took him 90 minutes to paddle back to school. What is Ethan s rate of paddling in miles per hour? How fast is the river current? 1 of 9
Suggested Grade Span 6 8 Task Ethan went on a kayaking expedition down the Ottauquechee River last week. He left school at 2:35 and paddled downstream 3 miles until he hit the sewage treatment plant at 3:05. He decided to get out of the river at that point (wise decision) and went across the street to Pizza Chef. He then headed back up the river in his boat. It took him 90 minutes to paddle back to school. What is Ethan s rate of paddling in miles per hour? How fast is the river current? Alternative Versions of Task More Accessible Version Ethan went on a kayaking expedition down the Ottauquechee River last week. He left school at 2:50 and paddled downstream 3 miles until he hit the sewage treatment plant at 3:10. He decided to get out of the river at that point (wise decision) and went across the street to Pizza Chef. He then headed back up the river in his boat. It took him 90 minutes to paddle back to school. How many times longer did it take him to paddle upstream? More Challenging Version Ethan went on a kayaking expedition down the Ottauquechee River last week. He paddled downstream at 10 miles per hour. He got out of the river, and went across the street to Pizza Chef and had lunch. He paddled back upstream at 8 mph. If his total paddling time was 27 minutes, how far did he paddle in all? Context This task was given to students while studying ratios and rates. They had previously used dimensional analysis to convert units of measurement. 2 of 9
What This Task Accomplishes This task allows students to see a real-life situation where dimensional analysis and rates can be used. The situation was one which students could identify with, as names and places were familiar to them. They were quickly engaged in the task. Time Required for Task Most students completed the task within the 45-minute class period. Interdisciplinary Links The students were able to use this problem as a refresher for doing science labs involving rate and dimensional analysis. Teaching Tips It is essential that students understand that a ratio is a comparison of two quantities by division, and a rate is a ratio that compares two different types of quantities (in this case miles and minutes). There should also be some previous work with current problems (river or air current), with discussion about what happens when you are traveling downstream (the river helps you go faster) and what happens when you are traveling upstream (the river works against you and slows you down). This problem could be modified for special needs students by changing the question to: What was Ethan s rate, in miles per hour, on his trip to Pizza Chef, and what was his rate on the trip back to school? Suggested Materials Provide students with calculators. Possible Solutions Ethan s rate of paddling is four miles per hour and the river current rate is two miles per hour. Most students will correctly identify the combined rates (downstream Ethan goes three miles in 30 minutes, upstream he goes three miles in 90 minutes) and, using dimensional analysis, will convert them to six miles per hour downstream and two miles per hour upstream. Depending upon skill level, the student will use either a system of algebraic equations or guess and test to solve for Ethan s rate and the river current s rate. 3 of 9
More Accessible Version Solution It took Ethan 20 minutes to paddle downstream. It took him 90 minutes to paddle back to school. 90 20 = 4.5 times longer More Challenging Version Solution 10x = 8[(27 60) - x] 10 8x = (27 60) - x Rate Time Distance Downstream 10 x 10x Upstream 8 (27 60) - x 8[(27 60) - x] (8 8)x + (10 8)x = 27 60 2.25x =.45 x =.2 10x = 2 miles x 2 (for round trip) = 4 miles in all Task-Specific Assessment Notes Novice The Novice will not have a strategy for finding the rates asked for in the task. S/he will most likely state that Ethan went three miles in 30 minutes downstream and three miles in 90 minutes upstream. The Novice will not be able to use dimensional analysis to convert those rates to miles per hour. Apprentice The Apprentice will have a strategy for solving part of the problem but will be unable to complete the task successfully. S/he will successfully find the rate in miles per hour for the trip downstream and upstream but will not have a strategy for continuing the task. Practitioner The Practitioner will have a strategy for successfully completing the task. The answers will be correct. S/he will most likely use guess and test to find the individual rates and will use appropriate and accurate math language to explain the solution. An accurate and appropriate math representation will also be included. 4 of 9
Expert The Expert will have a strategy for successfully completing the task, and the answers will be correct. S/he will assign variables for Ethan s rate and the rate of the current and use a system of algebraic equations to find those rates. These steps will be clearly explained with accurate and appropriate math language and representation. There will also be a relevant mathematical connection. 5 of 9
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