Exploring the power of Place Value

Imagine learning Math as if it were a foreign Language.

The English language has an alphabet of 26 letters. Two of those letters are used alone as words - “a” and “I”. We can’t even make a simple sentence with this two letters. Mimicking texting we could incorporate other letters that sound like words. “u”, “m”, “r”, “b”, “c”, “g”, “j”, “k”, “l”, “o”, “p”, “q”, “t”, and “y”. With these you could make some silly sentences such as the following:

O g, I c a b!

R u k?

Y?

I m.

To really have the ability to communicate you need to be able to combine these letters to form words and be able to master both conversation and reading. The average reading vocabulary for a 10-year-old may be 20,000 to 25,000 words.

Now consider basic elementary math as analogous to language learning. There are 10 “letters” that form the building blocks for math words: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Each of these letters is also an individual word. 0 students failed the test. 1 last brick to finish building the wall. We need 2 glasses of milk, etc.. (Of course if we were writing these sentences in normal sentences we would use the words zero, one, and two. But that is exactly the point - the numerals act like words and express meaning.)

Now the real power of our base 10 number system and of place value is the ability to expand a student's math vocabulary to literally trillions of words. Each of these numerals by themselves, as we said, may be used to indicate a simple number of 9 or less items. They also can be used to enumerate multiple groups of items. For example, the measurement 1 yard is equal to 36 inches - which is 3 tens and 6 ones. (In English we are somewhat handicapped because our students learn when counting to say “thirty-six”, whereas in some foreign languages students grow up counting with words in their language that would be the equivalent of saying, “three tens and six.”) Similarly, groups of a hundred are expressed as multiples up to 9 hundreds.

These groupings - hundreds, tens, and ones - can be replicated in further groupings of thousands, millions, billions, trillions, and beyond. With a little practice a student can read any of a trillion numbers!

Basic math facts covering addition and multiplication include one hundred facts for each operation. Since both addition and multiplication have the commutative property, 3 + 6 = 6 + 3, memorizing fifty facts gets a student well on the way to math facts fluency. Subtraction and division are the inverse of addition and multiplication, and are thus a little more complicated. Yet, if a student is fluid with addition facts a little effort will produce results in learning subtraction facts.

Now the really exciting part. 3 + 2 = 5. Wether it is ones, tens, or hundred trillions! The hardest part is when the addition or multiplication result in a sum or product greater than 9. Addition is the easiest since the largest sum of two numbers is 18. Every sum greater than 9 will result in a 1 in the next higher place value. 9 + 9 results in a 1 in the ten’s place and 8 in the one’s place. 9 ten trillion + 9 ten trillion will result in 1 in the hundred trillion’s place and an 8 in the ten trillions place. With multiplication the number that is in the next higher place could be anything from 1 to 8. When we carry the 1 it may result in 3 number addition. 1 + the first addend and then adding that sum to the second addend. It still won’t go above 1 in the next higher place value. For example 99 + 99: 9 0nes + 9 0nes = 18, 8 ones and carry the 1 ten. 1 ten + 9 tens = 10 tens or 1 hundred and 0 tens. 0 tens plus 9 tens = 9 tens. Which is a long winded explanation of why 99 + 99 = 198.

Multiplication is more complex because there are multiple columns to keep track of. For example: 99 X 99. One way to think of this problem is to stick to one numeral times one numeral. 9 ones time 9 ones = 8 tens and 1 one, 0r 9 X 9 = 81

9 ones time 9 tens = 8 hundreds and 1 ten, or 9 X 90 = 810

9 tens time 9 ones = 8 hundreds and 1 ten, or 90 X 9 = 810

9 tens time 9 tens = 81 hundreds or 8 thousands plus 1 hundred. or 90 X 90 = 8,100

Adding the different places: 8 thousands plus 8 hundreds pus 8 hundreds plus1 hundred, plus 8 tens plus 1 tens plus 1 tens, plus 1, or: 9, 801

Or looked at another way:

9 X 9 = 81

9 X 90 = 810

90 X 9 = 810

90 X 90 = 8,100

9,801

The key here is to realize that if a student knows that 9 times 9 = 81, it is only a matter of understanding how place value works with simple math facts to arrive at the correct answer.