Using masking or other such tape, the paper clips that form the number chain can be attached to each other at their ends, to form relatively rigid groups, each of which consists of the same number of paper clips. These groups, consisting of the same number of clips, still are linked to each other forming a chain of groups. Using such a chain the students can now count by multiple such as counting byetc. (These chains are similar to the Montessori’s bead-chains.) If we wish to have a compete set of chains, covering all the numbers from2, 3, 4,through1, counting up to12on each chain20approximately,paper clips, all of the same size are needed.6,500Note, however, that since the beads used by the Montessori chains are spherical, it is easy to use them to construct squares and cubes. Paper clips cannot be used to form squares and cubes. On the other hand, snap cubes are ideal for these tasks.

Linear MultiplicationFigure 6. Linear Multiplicationwith a Paper-Clip Number Chain

(animate this image)

Linear Multiplication in the Plane## Figure 7.

(animate this image)Linear Multiplication in the Planewith a Paper-Clip Number Chain

## Experiment, observe & log — fixed-length rectangular perimeter variable area using a paper-clip chain.

Areas Enclosed within a Fixed-Length Rectangular Perimeter

A 24-link chain.Example.

Instructions:Prepare and handout to each student one paper-clip chain loops, each constructed of 24 identical clips.

Notice that the chain must have even number of clips to create a rectangle.Working inthe Students set different rectangles, measure and calculate the area and perimeter of each rectangle and log their findings.pairs

Figure 8.Variable-area Rectangle from Paper-clip Chain Loop(animate this image)

Table 1.The Areas of Rectangles Enclosed by a 24-link Chain## As the name implies, we use the

Introducing theMeasuringNumbersnumber tomeasuringmeasureobjects. The number chain lets us measure the sizes of objects, along the three spatial dimensions, that is length, width and height. Unlike counting objects, we quickly discover that when we measure objects we often find that the size of the object does not always coincides with an exact count of the measuring unit. For example, the width of a book may bepaper clips and a part of the1213clip.^{th}

Back to why we labeled each clip at its end.We now have an opportunity to explain again why we placed the labels with the numerical names at the end of each paper clip. For if we labeled them in the beginning, the13paper clip, in the example above (measuring the width of a book), would have been labeled. The fact that it is not, indicates that only part of it can be included in the measurement, not the whole of it.^{th}

The birthday example.In case some students still have difficulty with this nuance of counting, you can use another example that your students may have better understanding. Point out that Alice’s eighth birthday party is celebrated at the end of a Alice’s eighth year and only then Alice is eight-year old.Make a paper-clip chain out of

paper clips, grouping every96clips together. Each clip represents a month and each group of12represents one year.12The measuring numbers are formally known as the fractions.

## Figure 9.

Fractions Created by the Number Chain(animate this image)

- The

- We’ve used chain links, whether they were made of paper
clips, snap cubes or some other objects of identical size and shape, to
create the counting and measuring numbers. Since the estimating numbers
cannot be known precisely, they cannot be measured and therefore, it is
impossible to use any number of repeating unit to create one. But this
is an excellent opportunity for a paper-clip chain to demonstrate its power.
Consider the following example.

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First we use our practice to use multiple chain clips to create the unitAs we recall, when we do this, each paper clip represents the unit fraction that equals the inverse of the number of paper clips that compose the unit1.In this example we select1.clips to represent the number10so each paper clip represents1,^{1}Now we can proceed and create a right-angle triangle, each of which having base and height of length/_{10}.or consisting of1paper clips. Now we form the hypotenuse using the remaining paper-clips (make sure the chain is sufficiently long.) We can see that the hypotenuse does not measure an exact count of paper clips. Its length is a little more than10clips, representing14and1^{4}or/_{10,}This is our first estimate. If we used1.4.paper clips to represent the number100, the length of the hypotenuse will be more than1clips, representing141and1^{41}, or/_{100}and so forth.1.41,

Note that in this case we exploit the inherent imprecision of chain links like paper clips, which is an advantage, while the precisely-interlocking snap cube cannot be used for this demonstration.

So it is not possible to use the number chain to represent any estimating (irrational) number, but as the name implies, we can use the number chain to estimate these numbers. And, in fact, we never need to know precisely the magnitude of any irrational world. In real-world applications it is usually sufficient to calculate the value of these numbers to the fourth decimal digit.

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