Tuesday, May 16, 2017

Unschooling Math: Binary Addition

Faced with the specter of having to memorize addition tables, and with the reward of building a calculator from scratch, our six year old—aka No. 1—and I have been working on math from a slightly different tack.  We switched to base 2 numbers.  Base 2 numbers, also known as binary, are the numbers all computers use.  For those unfamiliar with binary numbers, the binary system, (technically referred to as a ‘base 2’), only gives you two numbers to work with: 0 and 1.  Consequently, the binary addition table is far easier to memorize:


Addition Table
+
0
1
0
0
1
1
1
10


In contrast, the number system we’re all familiar with, (known as ‘base 10’), gives us 10 numbers to work with: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.  Given a single digit in our ‘normal’ base 10 system, we can represent up to 9 things.  If we have ten things, we have to add a new digit—known as the ten’s place—hence 10 uses two digits.  In base 2, given one digit, we can represent at most one thing.  So, when we want to represent two things, we have to add a new digit—known as the two’s place—so in the table above, when we add one with one, the result—two—is written as 10 in base 2.

The concept of ‘carrying’ was easier for us because we didn’t have to use such large numbers to practice.  You might not think adding 4 to 6 is a big deal, but you also might have memorized your addition tables more than a decade ago. 


No. 1 and I discovered that we when needed to carry what had really happened was that we’d run out of room adding two digits together.  In other words, when we add two one digit numbers together, and need to write the answer as a two digit number, we’ve carried.  For normal base 10 numbers, you ‘run out of room’ when you add two numbers and wind up with an answer larger than 9.  In base 2, you carry when you add two numbers and come up with an answer larger than one.  Consequently, we wind up carrying in almost every math problem.  No. 1’s getting all the benefit of practicing carrying without having to memorize a 100 entry addition table first.

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