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Q&A #326 |
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Dear Caron, I would like to share a lesson I teach to my students about the concepts of mean, median, and mode. I believe that students must first understand the concepts of data, and data handling, before applying them. I begin with a hands-on exploration of data without numbers. I pull out a board about .5 meter in length, and a handful of marbles (assorted sizes and colors). With saying a word I let the students figure out the connection between the board and the marbles. They determine my key question: how far will a marble roll off an inclined plane? (note: this is also a great physical science lesson). Students predict what they think will happen, and then we roll the marbles. Imagine looking down at the floor of my classroom. The group of marbles (I start with seven marbles) are now all rolled out from the board. I tell my students that this is really what data look like. I now have seven different answers to our one question. Our task is to take seven answers and use them to answer the key question. RANGE: I have a student put one of his or her hands in front of the marble that rolled the farthest, and the other hand behind the marble that rolled the shortest distance. This student then stands up and shows the class the range (the difference between the greatest and least data points). MEDIAN: Marbles do something special that numbers do not. They put themselves in order from greatest to least. This allow us to find the median (or middle) data point. I ask students to indicate which marble is the median. They easily can see that the median has three data points that have a greater distance, and three data points that have less distance than the median. (NOTE: they also see that the median is not exactly halfway between the greatest and least data points. Medians are the middle data point, not in the middle of the range). It is a great inquiry for students to determine how to find the median if another marble were added (and even number of data points). MODE: Are there some marbles that have rolled the same (or nearly the same) distance from the inclined plane? If there is some distance that was rolled more often than any other distance, this is the mode. (NOTE: you won't always get a mode.) MEAN: Did all seven marbles roll a fair share of distance from the inclined plane? Obviously, they didn't. A mean average takes the seven marbles and gives them each a fair share of distance. I use the "Robin Hood" method of determining a mean. Robin Hood was famous for "stealing from the rich, and giving to the poor". We then determine which marble is the richest (greatest distance) and poorest (least distance). Carefully, we move them towards the middle at the same rate. Keep doing this with all seven marbles until all marbles are the same distance from the inclined plane. The process of finding a mean is taking unfair shares and making fair shares. I then give each group of students a board and seven marbles. Their task is to use their marbles to re-answer our key question. When they have practiced it, and can explain their solution, they call me over to demonstrate what they can do with central tendency. Extension: at what angle must the inclined plane be set at to get the maximum distance of roll on the floor? I hope this makes sense. My student's really enjoy this, and the concept of data makes sense to them. Good luck, Paul Agranoff
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