Illuminations: Chairs

# Chairs

You walk into a restaurant with a large group of friends. To accommodate you, the restaurant will push a bunch of tables together—but how many tables will be needed to ensure that everyone has a seat? This applet will let you explore that question.

### Instructions

 The menus at the top of the applet allow investigation in a variety of ways. Mode: Allows the user to select Exploration, in which the number of chairs needed for a particular arrangement is displayed; or Guess, in which the user is able to construct an arrangement and then predict the number of chairs. Table: The applet allows two types of tables, 1 × 1 tables which accommodate one chair per side, and 2 × 1 tables, which hold two chairs on the longer sides. Arrangement: The user can choose a straight line arrangement (tables are placed side-by-side) or a rectangular arrangement (tables are placed along the border of a rectangle). The Increase/Decrease buttons allow the size of the arrangement to change.

### Exploration

 Set the menus at the top to Exploration, Six Chairs, and Rectangle. Using only the Length buttons, adjust the size of the arrangement. How do the number of chairs change with each increase in length? Using only the Width buttons, adjust the size of the arrangement. How do the number of chairs change with each increase in width? What pattern exists between the number of tables and the number of chairs? Describe a rule for the number of chairs based on what you observed. How many chairs are needed for a 10 × 5 arrangement? How many chairs are needed for a 13 × 8 arrangement? How many chairs are needed for an m × n arrangement?

### Lessons

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