Visualizing Place Value 2

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We mentioned that there are several visual ways of presenting the place value idea are available proceeding from the concrete to abstract. We presented 2 most concrete methods using bundles &amp; sticks and sheets, strips &amp; pieces. Let us now see other methods which can help children to proceed from the concrete to the abstract representations.

Essential Concept Underlying Place Value Manipulatives

The most important concept to understand is the relation between the various materials.

They are all in the ratio of the "base" of the number system which we use. The number system that we use in daily life has a Base of Ten.

This means the Flat is equivalent to ten Longs and the Long is equivalent to ten Pieces. Looked at in reverse, a Piece is one tenth of a Long and the Long is one tenth of the Flat.

For representing 4-digit numbers, we can think of a Cube which is equivalent to ten Flats.

In any system using a Base other than ten, we just have to replace the ten in the above sentences with the base of that system.

Manipulatives for representing a number in the Base 8 or Octal System will be scaled by 8!

The other important concept is that 1 can be represented by any of the manipulatives, as per our need.

Hence if we have an arrangement of 1 Flat, 2 Longs & 3 Pieces -

It will represent 123 (One Hundred & Twenty Three) if we take the Piece as representing 1.

It will represent 1230 (One thousand Two Hundred Thirty) is we take the Piece as representing 10.

It will represent 12.3 (Twelve point Three) is we take the Long as 1.

Bundles of Ten to just Bundles

Initially children should practice counting ten sticks and then grouping them into a bundle. Similarly they need to experience breaking up a bundle to count ten sticks.

Once they are familiar with the idea of bundles &amp; sticks, they can stop breaking up bundles by using the exchange idea. They can deposit a bundle into one &ldquo;bundle&rdquo; basket and take out ten sticks from another &ldquo;stick&rdquo; basket. Conversely, they can deposit ten sticks into the &ldquo;sticks&rdquo; basket and take a bundle from the &ldquo;bundle&ldquo; basket.

As the children get familiar with this idea they can move to another level of abstraction. The bundles (having actually ten sticks) can be replaced with a bundle (with any number of sticks) which is just identified by a rubber band around a few sticks.

Proportional Sheets, Strips &amp; Pieces to just shapes

The shift from bundles &amp; sticks to sheets, strips &amp; pieces is itself a step towards abstraction.

First, we can start with a strip which is 10 times a piece and a sheet which is 10 times a strip in size. They can even be cross hatched so as to bring this relation out.

Later the proportion can be abandoned as long as the Hundred and One are represented by a large square and a small square. The ten can be represented by a rectangle. Now a 3 digit number can be represented by a large squares &amp; small squares and long rectangles!

Shapes &amp; bundles to coloured tokens

At a later stage, the sheets, strips and pieces as well as the bundles can replaced by tokens which can either be coloured differently or have labels H, T &amp; O written on them.

Abacus

Here all tokens are of the same shape &amp; size. Their value depends of the location of the spike into which they are inserted. The spikes represent various place values and the number of tokens on the spikes represents the numerals which make up the number.

Numerals

From the abacus to numeral representation is a very simple one. The numerals represent the number of tokens at each place value.

Words

The word representation of numbers combines the difficulties of both math &amp; language. We will take this up in a subsequent Section (14) on Math & Language

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