Definition of Place Value

[jgrove]

What is the definition of place value?

Some background:
The term “place value” is ubiquitous in the standards. I’m wondering about language usage around the idea of place value and what exactly is meant by each term. Here are some ideas I have/have found:

place: location of a digit within a number (hundreds, tens, ones, etc.)

value of a place: This is referred to in the progressions: “In the base-ten system, the value of each place is 10 times the value of the place to its immediate right.”

place value: I have seen “place value” used in three different ways:
– The value a digit has by virtue of its position of a number (e.g. the value of 6 is 6, but the place value of 6 in 642 is 600). If this is the case, is “value represented by a digit” (used in 5.NBT.1 and 4.NBT.1) interchangeable with “place value”?
– The value of a place or position within a number (e.g. the place value of the ones place is 1). If this is the case, then “value represented by a digit” is not interchangeable with place value.
– As an umbrella term referring to properties and consequences of a base 10 number system.

Hoping to get some clarity to these discussions!

[lhwalker]

I don’t know enough to define “place value” definitively here, but I have learned to see the importance in the standards. Place value connects with “like terms” in all of math. In two-digit arithmetic, place value makes the difference between 20+5=25 OR 20+5=70, because we add like terms: tens to tens and ones to ones. Otherwise students memorize “line up on the left or line up on the right” and get those rules confused later. That’s why working with tens boxes or something similar is crucial in the lower grades. With decimal numbers, students often believe 0.25 > 0.8 because they don’t understand place value very well. With fractions, students confuse “multiply straight across” with “add straight across” if they do not understand we can only add like terms. In algebra, 2x + 3y = 5xy if a student doesn’t understand we can only add like terms. Pedagogically, I favor using interactive presentation software like Notebook. I type 536 on top of a white rectangle behind which I hide 500, 300 and 6. I drag the hidden numbers out from behind the white rectangle so students can all clearly see the value of each digit. I have to do this for some kids who make it all the way to high school, still fuzzy about place value.

[Bill McCallum]

I think all occurrences of the term “place value” in the standards could be replaced by the term “place value notation” without changing the meaning. Place value notation (as I’m sure you know!) is a the system of writing numbers where, for whole numbers,

• Each number is represented as a sequence of digits 0–9.

Each digit is assigned a value equal to the digit times a power of 10, the power being 1 for the right most digit, then 10, 100, 1000, etc. as you move successively to the left.

• The number is the sum of the values of the digits.

For decimals the system is extended by putting a decimal point at the end and adding digits to the right of the decimal point, whose values are the digit times 1/10, 1/100, etc.

The place value system is this system of notation. So, “the value represented by a digit” is not synonymous with “place value” … rather it is determined by place value (notation).

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