Visualizing Factors with a Base 12 Chart

Purpose: Students will use patterns to visualize factors and multiples.
Background: This lesson was proposed for fifth or fourth grade, but was used in an urban sixth grade.

Concept:

 Charts of numbers will contain patterns other than the patterns on which they are based.
A chart can be made around any base, but you can easily find the most patterns with bases 6, 12, and 24. On a base 12 chart, 2,3,4,6,11,12, and 13 will create patterns that a very easy to see; 5, 8, 9, and 10 will create patterns that are almost as easy to see.

Method: Give students a base 12 chart, or have them make a base 12 chart.
(Creating a base 12 chart makes good basic spreadsheet practice for those who are starting basic computer skills.) Assign different students to shade in all the multiples of different numbers, and then describe the patterns that they see. They should compare their charts to those charts made by other students. How are the patterns similar; how are the patterns different?

Updated 10/3/2009

 

 

 

Worksheet in Excel
 

Example:

purple numbers = multiples of 13

green numbers = multiples of 11

blue backgrounds = multiples of 4

red backgrounds = multiples of 5

 

Questions and Extensions:

  1. How are the patterns for 3 & 4 similar?
  2. How are the patterns for 11 & 13 similar?
  3. What is the relationship between the patterns for 2, 4, and 8?
  4. Predict which numbers on a base 10 chart will have patterns that are similar to 4, 6, and 11. Make a base 10 chart to find out. Why do those numbers have similar patterns?
  5. Chose another base for a chart. Predict the patterns you will see. Make the chart. Try to determine what relationships between the numbers creates each pattern.
  6. Draw Pascal's Triangle. Chose a number and color in its multiples. What patterns do you see?

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Here's an example of another geometric arrangement of numbers with two patterns marked with color. If the first row is extended out to 43, you can mark one of the longest regular sequences of primes that is known.

Blue = perfect squares

red = a sequence of primes

Related Pages:
 

Super Extension:

Converting from fractions to "decimal"* in base 12 is much easier than in base 10. However it would be harder to convert fractions to "decimal"* in base 11. If students have had exposure to the idea of converting between bases, they should look at the following fractions in a few different bases to predict which base is easier for fractions: 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/12.

* Technically decimal means base ten. Perhaps for base twelve we should say "the duodecimal point", for base eleven "the unidecimal point," and for base six "the hex point."
 
     
 

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