Place Value

Other Rules

Let’s play the dots and boxes game, but change the rule.

The 1←3 Rule

Whenever there are three dots in single box, they “explode,” disappear, and become one dot in the box to the left.

 

Example: Fifteen dots in the 1←3 system

Here’s what happens with fifteen dots:

 

Solution: The 1←3 code for fifteen dots is: 120.

 

Problem 2

  1. Show that the 1←3 code for twenty dots is 202.
  2. What is the 1←3 code for thirteen dots?
  3. What is the 1←3 code for twenty-five dots?
  4. What number of dots has 1←3 code 1022?
  5. Is it possible for a collection of dots to have 1←3 code 2031? Explain your answer.

 

Problem 3

  1. Describe how the 1←4 rule would work.
  2. What is the 1←4 code for thirteen dots?

 

Problem 4

  1. What is the 1←5 code for the thirteen dots?
  2. What is the 1←5 code for five dots?

 

Problem 5

  1. What is the 1←9 code for thirteen dots?
  2. What is the 1←9 code for thirty dots?

 

Problem 6

  1. What is the 1←10 code for thirteen dots?
  2. What is the 1←10 code for thirty-seven dots?
  3. What is the 1←10 code for two hundred thirty-eight dots?
  4. What is the 1←10 code for five thousand eight hundred and thirty-three dots?

 

Think / Pair / Share

After you have worked on the problems on your own, compare your ideas with a partner.  Can you describe what’s going on in Problem 6 and why?

 

License

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Mathematics for Elementary Teachers by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.