Converting Base 2 Binary Numbers to Base 10 Decimal

Over the last few weeks we have been exploring Base number systems different from our own decimal Base 10, which we take for granted. Our system (which uses 10 symbols) is much less efficient than the ancient Mayan vigesimal Base 20 system that uses only 3 symbols (a dot for 1, a stick for 5, and a shell for 0). 

This week, i had the students imagine a system that was even more efficient than the Mayan system that used only two symbols (1 for ON and 0 for OFF). They were surprised to learn that this binary Base 2 system is used by computers to generate numbers, letters, and all commands. Yes, the hexadecimal system (used even more often to code computers) will be taught at the higher grade levels in the coming weeks.

As our system has place values that multiply by 10 (….. 10,000, 1000, 100, 10, 1), the Mayan system has place values that multiply by 20 (… 160,000, 8000, 400, 20, 1), the binary Base 2 system has place values that multiply by 2 (… 16, 8, 4, 2, 1). There are more similarities between the Base 20 and Base 2 system than with our decimal base 10 system.

In the pdf below, you will see an example page where I only wrote in a place value (… 16, 8, 4, 2, 1) if there was an “ON” symbol, the “1” and then added the values. For example, if the Base 2 number is 1010, reading the plane values from right to left, i wouldn’t write a value of 1 above the 0, i would write a 2 above the 1, then no 4 above the 0, and finally, an 8 above the left most 1. So i am simply adding the 8 + 2 to generate a base 10 number of 10.

For the K-1 students, 

category one

About three quarters of them could write the powers of two (… 16, 8, 4, 2, 1) above the on buttons and add them to for their base 10 equivalents.

category two

The rest were able to just write the powers of two (… 16, 8, 4, 2, 1) above the on buttons. 

category three

A few students who struggled to follow this pattern were just asked to practice their first few doubles using their fingers: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512 ….).

Students should only work any their level and not feel any pressure to master these concepts right away. We will revisit this language each year. My philosophy is that they understand that our number system is not the only one and place value is everything. Numbers take on so many forms and languages. The earlier they learn this, the more facility for numbers they will have.

For the 2nd through 5th grades, I created two different pdfs with challenges from 2 to 6 figures for 2-4th grades and up to 7 figures for 5th grade. 

For example, if the student sees 1 0 1 0 0 1 0 1 0 0, they should start on the right and put the place values of (… 16, 8, 4, 2, 1) above only those values with a “1.”

256  64       16     4

1 0 1 0 0 1 0 1 0 0 ,then add up the place values of 256  +  64 +16 + 4 = 340

The longer numbers are extremely challenging. The most difficult question on the 5th grade worksheet is 100101010010111010010101 in binary which equals 9,776,789 in decimal.