
Linear Arithmetic Blocks (LAB)
Our preferred model for decimals
Description of LAB
Linear Arithmetic Blocks  our concrete model of choice for
teaching decimals!
Numbers are represented by length of pieces of plastic pipe.
The longest piece, representing "one" is just over a metre
long. This piece is shown to the students first. Discuss
cutting into 10 equal pieces. Ask students to use their hands
to indicate how long the new piece will be, and what name it should
have (a tenth). 
Now show the students a tenth
piece and discuss cutting it into 10 equal pieces. Again, ask students
to predict the length of the new piece and a name (a hundredth). 
Now show the students the hundredth
piece and discuss cutting it into 10 equal pieces. Again, ask students
to predict the length of the new piece and a name (a thousandth).

Now show the students the thousandth
piece. We use washers for thousandths because it is too hard to cut
such thin pieces of plastic pipe. Then discuss cutting the washer
into 10 equal pieces, to make tenthousandths and beyond to smaller
and smaller pieces so that students understand the process that creates
the endless base ten chain. In practice, we make nothing smaller than
thousandths.

LAB pieces can be arranged randomly in piles or in 2 systematic
ways. On the right, they are shown on an organiser. The organiser
is a wooden stand with three dowel rods to hold nine tenths, hundredths
and thousandths. This is very useful to demonstrate trading in the
addition and subtraction algorithms. In the 3 photos below, the
pieces are laid end to end to confirm that:

10 tenths have the same length as the one piece


10 hundredths have the same length as the tenth piece


10 thousandths have the same length as the hundredth piece


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Features of LAB as a model

Uses the physical quantity of length to
represent the size of a number. 

Represents size of number from the digits in the numeral very well.


Represents base 10 properties (bundling and column overflow) very
well.


Useful from ones to thousandths and can be mentally extended easily
in both directions. (This is an important discussion to have with
students!)


The "endless base ten chain" multiplicative relations
between the values of places are shown reasonably well.


Demonstrates addition and subtraction algorithms well.


Demonstrates multiplication and division of a decimal by a small
whole number or by a power of ten well.


Only some divisions by a decimal can be shown well (selected quotitions).
This is a general limitation of concrete models.


Multiplication by a decimal cannot be
easily demonstrated with LAB. 

LAB represents numbers by the quantity
of length, not by units of length such as millimetres.
This is an important distinction because using units of length may
perpetuate misconceptions that the decimal point simply separates
one wholenumber quantity (the number of metres) from another (the
number of millimetres). It happens that the unit piece is approximately
one metre long, but this is only a consequence of the size of suitable
materials. 
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How to make LAB
Linear Arithmetic Blocks (LAB) can be made at home or at school
from ordinary washers and PVC pipe of a similar diameter. To make
LAB, first purchase the washers (thousandths). The next pieces (hundredths)
are then made by cutting small lengths of plastic tubing to exactly
to the length of 10 washers. Then make tenths (medium size lengths
of tubing ) cut to 10 hundredths. Lastly, the one (whole) is cut
to 10 tenths and is a rather long piece of tube, probably over a
metre long (as the washer may well be more than 1 mm thick). 

A good set for classroom demonstration requires about 7m of 25
mm diameter plastic pipe and contains
 about 40 thousandths
 about 30 tenths and hundredths
 at least two ones.
Children's sets can have fewer pieces.

The wooden organiser is optional. It is made of 3 rods (tenths,
hundredths and thousandths) set into a base of wood. A decimal point
can be drawn on the base with texta.
The heights of the rods on the organiser are such that only
9 of the pieces can be placed on the relevant rod. It is obviously
not practical to make a rod for the ones, as it would be over 9
metres high! (Involve students in this discussion)
Approximately 1.5 m of dowel is required and a base measuring approximately
30cm x 15cm x 10cm. 

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Using LAB as a model of the numberline
In the pictures below, the LAB pieces representing the decimals
have been laid linearly instead of on the organiser. This
allows us to compare the total length of the pieces and hence the
size of the decimals they represent.

LAB clearly shows that 0.2 is larger than 0.13 and not the other
way around! Many longerislarger children think that 0.13 is larger
than 0.2 (as 13 is larger than 2).
It also shows other properties clearly including:
 equivalence of 0.2 and 0.20 (2 tenths and 20 hundredths)
 equivalence of 0.13 (13 hundredths) with 0.1 + 0.03 (one tenth
and 3 hundredths)
 density of decimal numbers (that there are other decimals between
0.24 and 0.25 or between 0.247 and 0.248 etc)

LAB can be used to round decimal numbers. For example, to round
0.27 to the nearest tenth, make several numbers using only the tenths
pieces (1 tenth, 2 tenths, 3 tenths, 4 tenths...) and compare. Children
can see that the number 0.27 is between 2 tenths and 3 tenths and
closer to 0.3. 
Many children and even some adults
are confused by numberlines. Building a number line is the ideal way
to help students understand the ideas involved. 
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Using the organiser
The organiser serves a similar purpose to the place value chart
often used with MAB, holding pieces of the same size together and
representing the leftright spatial arrangement of decimal numeration.
(Beware  you must be looking from the front!)
It also if there are ten or more pieces of any one size, they will
not fit onto the appropriate rod and so ten of them must be exchanged
for a single piece of the next highest value. This forces the trading
required in the algorithms.


Overuse of the organiser or place value chart may reduce the LAB or MAB
model (in the children's minds) to that of an abacus; where it is the
position of the beads only which represents the size of the number.
To ensure that students do not become reliant on the organiser (or place
value chart), make sure that students:

appreciate that the size of the number represented does
depend on the physical length of the blocks and not their
arrangement on a chart, and


are able to move the blocks onto the organiser (by themselves)
to assist in reading off the symbolic representation.

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Why we prefer LAB to MAB
Our research has demonstrated that LAB has a number of advantages over
MAB.
Our published research paper (Stacey
et al, 2001a) compares the two materials on the basis of epistemic
fidelity (how true the model is to the mathematical principles involved)
and accessibility for students. Two teaching experiments involving 30
matched students indicated that LAB is considerably more accessible for
students. There are three reasons for this:

students get confused with MAB simply because it has been used
before with the "mini" representing one (see above);


LAB models number with length whereas MAB models number
with volume and many students in upper primary do not yet
have a strong grasp of volume;


the various pieces of MAB seem to be of different dimensions (1D,
2D, 3D ) and this makes generalizing to more place value columns
difficult.

Use of LAB was associated with more active engagement by students
and deeper discussion. Epistemic fidelity is critical to facilitate
teaching with the models, but we attribute the enhanced classroom environment
to the greater accessibility of the LAB material.
Both models have excellent epistemic fidelity, so that they both show
how the size of numbers depends on the digits and the place value columns
and they both can be used to demonstrate the operations. However, a significant
difference between the two models is that LAB, with pieces laid end to
end, has structural similarity to the number line, and is thus better
able than MAB to model number density (the property that between any two
decimals, a third decimal can always be inserted). LAB is therefore better
able to demonstrate the principles of rounding. The limitations of MAB
in regard to the continuous properties of decimals were noted by Hiebert,
Wearne and Taber (1991). Following instruction with MAB, Year 4 students'
performance on discretecontext tasks (e.g. writing the number represented
by a picture of MAB; choosing the larger of two decimals) improved. However
their performance on continuousrepresentation tasks (e.g. shading 2.6
of a continuous quantity; finding a number between two decimals) did not.
Thus MAB appeared to support understandings of decimals as discrete quantities
but not as continuous quantities.
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Sample lesson plan
LAB was first shown to us by Heather McCarthy, a Melbourne teacher. Heather
had seen it at an inservice day, but we cannot trace the source.
Heather's
worksheets.
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