We spent several weeks studying the Mayan vegidecimal (Base 20) number system with 20 symbols, then the binary (Base 2) system used by computers with only 1s and 0s, and last week, used the binary system to look at exponential decay using 1/2, 1/4, 1/8 and so on. 

 

This week, we continued this journey through alternative bases by studying Base 16 or hexadecimal numbers used to program computers. The place values are a geometric sequence of x16 (… 65,536, 4,096, 256, 16, 1). Since we need 16 symbols taking up only one character, we have a problem after 0,1,2,3,4,5,6,7,8,and 9, because 10-15 has two digits. So we use A=10, B=11, C=12, D=13, E=14, F=15. 

 

For example, the base 16 number A2 means 10 x 16 + 2 x 1 = 162 in decimal base 10. The base 16 number 2B5 means 2 x 256 + 11 x 16 + 5 x 1 = 693. The K-1 group looked at base 16 numbers with one and two digits in the 16s place and the 1s place. The 2-3rd graders used base 16 numbers in three place values of 256s, 16s, and 1s. The 3-5th graders used base 16 numbers with 4-7 place values.

 

I used every word I could think of using the letters A-F. Two letter words included ED, BE, AD, and other abbreviations. Three letter words included ACE, BAD, DAD, DAB, CAB, CAD, FAB, FAD. Four letter words included CAFE, DEAF, DEAD, FACE, FADE, BEAD, BEEF, FEED, DEED, and beyond that: DECAF, BEADED, DECADE, DEFACE, FACADE, DEFACED, and then I made up some compound words for those very ambitious 5th graders such as BEEFCAFE for those non-vegetarians. This 8 letter word’s first letter B is 11 x  268,435,456. 

 

The strategy I employed was to require the children to start with 10 of any number so if they have a letter C 16s, they need 12 x 16. Start with 10 x 16 which is 160 and then add two more 16s or 32. If they need D x 256, start with 2560 and then add three more 256s by adding 512 + 256 to 2560. When using F=15 of any place value start with 10 by tacking on a 0 and then adding half of that number since 5 is half of 10.

 

After working hard this week, the children should be somewhat literate in the base 16 language.