CTC II Summary 1/14/2013

A Whole Lotta Popcorn!

So, the question remains:  Just how much popcorn would it take to fill our classroom? 

Students continued their analysis and discussion of the "popcorn question" in class this week.  Our two groups came together to compare notes on their independent measurements of room size and the overall space that the popcorn will need to occupy.  Having taken measurements of the room dimensions in a previous class, students worked on subtracting out the space occupied by certain immovable components such as cabinets, shelves, and ceiling soffits before computing an overall room volume.  One important objective of the project is to observe how similar (or dissimilar!) the approaches, measurements, and final estimates are between two groups working independently.  The process has provided some basic, worthwhile lessons so far:

• Keep organized, detailed notes!  Students scrambling to find and then decipher handwritten notes and measurements quickly realized the importance of carefully recording legible measurements and notes.  The old carpenters adage "measure twice, cut once" applies here too.  And we might add to that, "organize, label, organize!"
  
• Determine an appropriate measurement unit.  One group chose to record room measurements in feet while the other group worked in units of inches, so conversions were needed before the two group measurements could be compared.  This provided all students the opportunity to think through how to convert cubic inches to cubic feet (1 cubic ft = 12 in x 12 in x 12 in, or a conversion factor of 1728). Although either unit is acceptable, it makes more sense to measure a classroom in feet rather than inches.  Thinking through the appropriate unit for the scale of a problem is an important step one decision.

        It turns out that even after converting to a common unit, our two groups came up with quite disparate         estimates of volume for the classroom.  We need to investigate further to determine why this occurred. 

• Verify your estimate.  How will students ultimately determine whether their popcorn estimates are "right"?  Is it possible to determine a final answer without actually filling the room with popcorn?  Students were asked to consider these questions and come up with an approach to verify their estimates. What unit should be used for popcorn measurement - number of kernels, weight, bags of popcorn? 
  

        We will hope to answer these questions next week as we finalize our estimates and perform tests and             verifications using real popcorn. Yum!  Stay tuned....

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The Value of Knowing How to Estimate

For students, the question may arise at this point - exactly why should we care how much popcorn fills the classroom?  Well, I'm glad you asked!  It turns out that many seemingly inaccessible questions such as these arise in our everyday, academic, and professional lives (or will - somewhere down the road), and therefore cultivating a working knowledge of how to make sound estimates is an important aspect of what is required to become a good problem solver.

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How much money will I need to save to pay for a full college education? How many bricks would be needed to construct a building of a preferred dimension?  How much breathable air is contained inside a sealed room of a given size?  How many people could survive inside and for how long?  How much agricultural land is required to feed a family of 5?  100 people?  a nation?  If everyone in the country needed to be inoculated against a virulent strain of the flu, how quickly could this be accomplished?  These and an infinite number of other varied questions and problems require us to make well-informed decisions based on a chain of reasoning, estimation, and calculation which require skills you are asked to use in the popcorn problem:

• ability to simplify a problem into its component parts
• determination of appropriate units and accurate measurement
  
• identification and collection of relevant data
  
• ability to evaluate many potential approaches and solutions
  
• synthesis of knowledge to make educated guesses
  
• making decisions as part of a group
• communicating mathematical ideas verbally and in writing
• verifying conclusions
• identifying sources of error and bounding the problem space

So, while we may never actually need to fill a classroom full of popcorn (as fun as that would be!), students should recognize that the process of determining how we might do so develops and broadens our critical thinking and problem solving abilities in a variety of ways that truly are relevant and valuable.

See you all next week!