Counting+the+fish

Lesson description:
Note: This is a simple description of the lesson. Many details have been omitted.

Students are shown a short video clips in featuring animals in various sized packs. Students are asked to count how many animals are on screen at any given time. The task is easy with small number. A school of fish or flock of birds presents a problem. Estimating techniques have been given by students. Challenge is reiterated with school of fish (3 dimensional, uneven distribution,... ). Students are introduced to scientific way of counting population: capture and release. They are then given bags of green marbles that are to serve as fish in a lake, and a smaller bag of blue marbles that are to serve as tagged fish. Students perform experiments and discuss their findings.

Second day the students are presented with a second problem. They are given a bag of m&m's and are asked to count how many candy are in the bag without actually counting the candy.

Students' work involved:

 * revamping experimental design (need to change sample size,...)
 * estimating rations; percents
 * setting up proportions
 * estimating quantities
 * finding median/mode
 * noticing discrepancies -- wide range of answers
 * discussing decimals that appeared as answers to "how many fish?"
 * discussing precision vs. accuracy
 * different ways of counting -- not necessary to do capture and release

Value of problem posed:

 * multiple access points
 * real world use/not contrived
 * not just reading/hands on -- relieves boredom
 * picture describes proportion
 * talking about why things happened
 * talking about extremes leads to conversations about statistical and other topics
 * led to students' own questions
 * incorporates technology
 * provides "anchor" experience - common experience you can refer back to
 * it can be used at different levels: serves multiple purposes

Questions that need to be addressed:

 * 1) Where do these two different problems fit into the "continuum" of teaching with problems?
 * We can add the third problem: Give students a bag of green marbles and ask them to come up with a reliable way of estimating the number of marbles without counting.

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