Place Value - Development for age 5 - 9+ Concepts, Activities, Assessment, and Evaluation
No place value understanding. Children see ten as the same kind of number as 1,2,3, ... 9, only bigger. Can count to ten and say they have ten fingers or ten toes. Most likely do not conserve or have cardinality.
- Ten is a bigger number than 1, 2, 3, ... 9.
- Ten is less than 11, 12, ....
Children can use one-to-one relationships and systematically count objects accurately to at least twenty. They can also skip count by tens and understand skip counting as repeated addition. Not multiples of ten. There is no place value understanding. Students understand numbers can be grouped with different combinations and ten is just one of several groupings. Students are easily confused by the one in teens and two in twenties as one and two rather than ten or twenty.
When asked they will represent numbers greater than ten with a one-to-one relationship For example: showing how many students in their class by drawing designs so one design represents one student. Not groups of tens and ones. They are working toward unitizing (counting groups of objects in equivalent groups using a one-to-one strategy; like counting five groups of ten by 1, 2, 3, 4, 5 and recognizing it as five equal groups of ten or 50) but have not achieved unitizing.
- Objects can be grouped in tens.
- Grouping by tens makes it easier to count.
- Objects can be grouped into tens and leftovers.
Place value models should be actual objects put into groups of ten.
Students can inventory books in the classroom by creating stacks of ten books and can recognize the groups of ten as units as well as a group of ten, but not simultaneously. Total books as groups of ten and extras.
Place value models need to be proportional for children.
At his level students understand Number value - cardinality to 100, can order and sequence numbers to 100, know zero is the absence of objects, understand hierarchical use of numbers in a counting sequence, and more than one addition fact relate to a number's value (but not hierarchical inclusion of addtion related to number sense), and those facts can be used to compose and decompose numbers.
- Ten is a group of ten ones.
- Objects in groups of ten can be combined by skip counting by tens.
- Counting by tens is helpful.
- Counting by tens is like counting by ones with zeros.
- Objects can be grouped into tens when thinking of number value and to solve problems. (significance of ten)
- Numbers can be grouped as ten and more.
While students can group objects by tens, they consider it as an additive process the same as putting objects into other groupings (pairs, fives, eight, four...) together.
Solves addition and subtraction problems mentally by decomposing into groups of tens and ones, composing the tens, then the ones, and adding the tens and ones.
May or may not know how zero is used to mark positions with no value.
- Unitize - groups of ten are simultaneously both: equal groups of ten and the equivalent number of units or ones. For example thirty is simultaneously three equal groups of ten and thirty ones. In 30 the three means three groups of ten or thirty ones (plus zero additional ones represented by the zero). In 32 the three means three groups of ten or thirty ones plus two additional ones represented by the two.
Students can count each group as one unit (1 of 10, 2 of 10, 3 of 10 ...) and realize that each unit has and must have an equivalent number of objects in it.
Students know that five groups of ten is fifty. They don't need to count by tens five times to know five groups is 50.
Students understand that groups can be regrouped as other equivalent groups (One group of ten is also two groups of five, and other combinations to make ten.).
Before students develop place value relationships they must unitize and usually have number value to at least 1000. Can round numbers to multiples of ten and can count forward and backward by tens and hundreds starting with any number. Knows multiples of ten including 10 X 10, and maybe 10 X 100.
Students who understand place value relationships know:
- Place value unitizes by tens, hundreds, thousands, ...
- Numbers can be locate to a nearest numbers with a value of ten (rounding).
- Ten has a significant role in our base ten number system. Ten, hundred, and thousand are important multiples of this number system.
- A number system is a code that uses a set of symbols (numerals) and rules for combining the symbols to represent number values.
- All numbers can be represented with only ten symbols or numerals.
- Ten numbers can represent any unit (ones, tens, hundreds, …).
- Zero has an important function in writing numbers.
- Position of a digit can represent different values.
- The sum of the value of all digits determines the value of a numeral (cardinality) The total value of a numeral is the sum of all digits values in its particular place. A digits value is a function of its number value and it's place value.
- Numbers can be composed and decomposed into different equivalent groups of tens and more (taken apart (decomposed) and put together in different order (composed) or regrouped and written in expanded notation.
- Numbers of groups can be operated on without regard to the value of the group (62 - 41), (60 - 40) can be operated as (6 - 4) and (2 - 1) and write, think, or say 21 without really (multiplying by ten). This thinking about and working with groups is a result of unitizing.
- Addition facts can often use ten as an anchor (32 = 30 +2) for addition and subtraction. This is helpful when adding (27 + 35), (27 + 30) + 5 = (57 + 3) + 2 = 62 both for adding or subtracting to make tens, and adding and subtracting in leaps of ten.
- All place values are multiples of 10 (10, 100, 1000, ...)
Unitizing place values of decimal numbers less than one: tenths, hundredths, thousandths ...
- Place value is exponential: 100,101,102, 103...
(50 = 5 * 101+0*100,
51 = 5 * 101+1*100,
52 = 5 * 101+2*100,
200= 2*102 +0*101 + 0*100,
201=2*102+0*101 + 1*100,
- Decimals as exponential: 10-1,10-2, 10-3
- Unitizing place values of decimal numbers, less than and greater than one, as an exponential progressions.
- Hindus are credited with the invention of a place value system.
- The Arabs are credited with applying and spreading its use.
- Translations of Fibonacci's work introduced it to Europe.