Mathematics Developmental Continuum P-10

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Linear Arithmetic Blocks

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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).

 Student with one metre plastic pipe
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). Plastic pipe broken into a tenth
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). Plastic pipe broken into a hundreth
Now show the students the thousandth piece. 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 ten-thousandths 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. Plastic pipe broken into a thousandth

LAB pieces can be arranged randomly in piles or in a long line. In the photos below, the pieces are laid end to end to confirm that:

10 tenths have the same length as the one piece

Students comparing whole pipe with 10 tenths of pipe
10 hundredths have the same length as the tenth piece

Students comparing tenth of plastic pipe with 10 hundreths
10 thousandths have the same length as the hundredth piece

Student comparing a hundreth of a pipe with 10 thousandths

 

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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 whole-number 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 match 10 hundredths. Lastly, the one (whole) is cut to match 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). Note that it is very important that you do not introduce the materials to the students in this order, as they will automatically assign the value of 1 to the washer and 1000 to the longest piece. Also note that you should not mark any of the pieces, (for example mark the one with lines to show it is the same as 10 tenths) as then the one rod will stop looking like a 1, and look like a 10.

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.

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Using LAB as a model of the number line

In the pictures below, the LAB pieces representing the decimals have been laid linearly instead of in a random pile. 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 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)

 

 Student comparing two 10ths of plastic pipe with 13 hundreths

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.

 Students comparing various sized sections of plastic pipe
Many children and even some adults are confused by number lines. Building a number line is the ideal way to help students understand the important ideas involved.

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Why we prefer LAB to MAB

Research conducted at the University of Melbourne (Stacey et al , 2001) has demonstrated that LAB has a number of advantages over MAB. 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;
  • 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 (1-D, 2-D, 3-D ) and this makes generalising to more place value columns difficult.

Use of LAB was associated with more active engagement by students and deeper discussion. Both models show how the size of numbers depends on the digits and the place value columns and they both can be used to demonstrate various operations with decimals. However, a significant difference between the two models is that LAB, with pieces laid end to end, has structural similarity to the number line so is 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.

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References

This page is published with permission from Steinle, V., Stacey, K. & Chambers, D. (2006) Teaching and Learning about Decimals. (Version 3.1 ) Faculty of Education, University of Melbourne . (CD-ROM). See also http://extranet.edfac.unimelb.edu.au/DSME/decimals

Stacey, K., Helme, S., Archer, S., & Condon, C. (2001). The effect of epistemic fidelity and accessibility on teaching with physical materials: A comparison of two models for teaching decimal numeration. Educational Studies in Mathematics, 47, 199-221.