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There aren't many places in the real world that Stem and Leaf Plots are used. However, the good news is the places they are used are very important.
These are 3 Extremely Realistic Ways to Introduce Stem and Leaf Plots
Airports  Everyone loves going on vacation. It would be awful to go to an airport to leave for vacation and not know how to read the flight times. Create with them a Stem and Leaf Plots showing flight departure and arrival times.
Subway Stations  Big cities are scary enough as they are. Imagine not knowing which train to get on at the station? If your students aren't from a big city then this will intrigue them. If they are from a big city this is a life skill they probably must have!
Comparing Athletes Stats  When trying to decide who the MVP should be for the NBA this season we can use a Back to Back Stem and Leaf Plot to compare their statistics. Simply put one player on each side, the first digit of the stat in the middle and the second digit as the leaves. I like to use these game by game stats. Point scored (there are 82 games so you will have sufficient data), Rebounds, Assists, Blocks, Steals, Minutes Played, and Turnovers (Make this red so they know it's a bad thing).
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