Place Value Base Ten Place Value The baseten

Place Value Base Ten

Place Value The base-ten place-value system is the way that we communicate and represent anything that we do with whole numbers and later with decimals.

Place Value Our place-value system, which allows us to represent any number with just 10 digits, develops across the elementary and middle grades. Kindergarten and 1 st grade – count and are exposed to numbers to 100. They begin to think about groups of ten objects as a unit. Grade 2 – these initial ideas of patterns and groups of ten are formally connected to our place-value system of numeration. Grades 3 and 4 – extend their understanding of numbers up to 10, 000 Grades 4 -5 – ideas extended to decimals

Pre-Base-Ten Understandings Kindergarteners can and should learn to count to 100 and count out sets of 20 to 30 objects. Their understanding is based on a one-more-than or count-by-ones approach to quantity. They are usually not able to separate numbers into place-value groups. Their pre-base-ten understanding is a “unitary” understanding, as they rely on unitary counts to understand quantities.

Basic Ideas of Place Value “Unitary” counting – counting by ones

Basic Ideas of Place Value Base Ten – counting by groups and singles

Basic Ideas of Place Value Equivalent Grouping – regrouping in addition and subtraction are based on equivalent grouping

The Role of Counting plays a key role in constructing base-ten ideas about quantity and connecting these concepts to symbols and oral names for numbers. The objective is to help students integrate the grouping-by –tens concepts with what they know about numbers from counting by ones.

Integration of Groupings with Words The way we say a number must be connected with the grouping-by-tens concept. § § § 2 tens and 3 ones 2 tens and 3 singles Each is correct, but it is best to select a way and use it consistently at first, especially if you have ELL students. The oral and written names of numbers should be developed at the same time as the conceptual understanding in addressed.

Integrating Groupings with Place-Value Notation The symbolic scheme we use for writing numbers, ones on the right and tens on the left, is coordinated with how we read from right to left. Tens Ones

3 Stages of Place-Value Development Concrete Semi-concrete Abstract/Symbolic Tens Ones 2 3 17

Base-Ten Models Physical models play a key role in helping students develop the idea of “a ten” as both a single unit as well as a set of ten units. Physical models can be categorized as § Groupable or § Pre-grouped

Groupable Models that most clearly reflect the relationships of ones, tens, and hundreds are those that for which the ten can actually be made or grouped from the single pieces. A proportional model

Pre-grouped Models Pre-grouped models cannot be taken apart or put together. A proportional model

Non-proportional Models Money is an example of a non-proportional model for base-ten.

Developing Base-Ten Concepts Because students come to their development of baseten concepts with a count-by-ones idea of number you must begin there. You cannot impose counting by ten on students. They need to experiment with large amounts in groups of like size and come to an understanding that ten is a very useful size to use. Treat the concept of 100 in the same way.

Developing Base-Ten Concepts Use activities that will incorporate oral language with equivalent representation ideas. Base-ten riddles are one possible way to do this. I have 23 ones and 4 tens. Who am I? § I have 4 hundreds, 12 tens, and 6 ones. Who am I? § I am 45. I have 25 ones. How many tens do I have? § If you put 3 more tens with me. I would be 115? How many tens do I have? Have students write their own riddles using base-ten words. §

Oral and Written Names for Numbers The ways we say and write numbers are conventions rather than concepts. Students learn these conventions by being told rather than by problem-solving. For ELL students the convention in our English number words is probably different that in their native language. This is especially true of the numbers 11 -19.

Numbers Beyond 999 By the end of second grade should understand number concepts to 1000. In third grade they extend their understanding to 10, 000. By grade four they learn place value to 100, 000. By fifth grade they should understand numbers in the millions.