# Logic — the Art of Reasoning

Mathematics  — the Art of Studying Patterns Using Logic

# The Number Chain (cont'd)

### Editorial Note

This document is a work in progress. At this time it contains only the introduction to the number chain and very few activities. Additional information and activities can be found in other Teaching documents of MathVentures.

## Multiplication with the Number Chain

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 by 2, 3, 4, etc. (These chains are similar to the Montessori’s bead-chains.) If we wish to have a compete set of chains, covering all the numbers from 1 through 12, counting up to 20 on each chain, approximately 6,500 paper clips, all of the same size are needed.

Note, 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 Multiplication

Figure 6.  Linear Multiplication with a Paper-Clip Number Chain
(animate this image)

# Areas Enclosed within a Fixed-Length Rectangular Perimeter

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

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 in pairs the Students set different rectangles, measure and calculate the area and perimeter of each rectangle and log their findings.

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

# Introducing the Measuring Numbers

As the name implies, we use the measuring number to measure objects. 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 be 12 paper clips and a part of the 13th clip.

• 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, the 13th paper 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.
• 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 96 paper clips, grouping every 12 clips together. Each clip represents a month and each group of 12 represents one year.
• The measuring numbers are formally known as the fractions.

•

# Introducing the Estimating Numbers

The Estimating Numbers are the numbers that we can never know precisely. Formally the estimating numbers are called the irrational numbers. This is the last group of numbers that resides on the number line. Together all the numbers that compose the number line are called the real numbers.

# How Can a Chain Links Create an Estimating Number?

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.

## Creating  From the Number Chain

We know that  cannot be expressed as a fraction. It is an estimating (irrational) number. (for the proof see The Square Root of 2 is Irrational. [This document is yet to be placed on the web.]) We will resort for the same geometric construction, which the Pythagorean used some 2,500 years ago when they first discovered that the is in fact an irrational number, to estimate the size of . But instead of using plane (Euclidean) geometry as they did, we will use the number chain.

# Figure 10.   Estimating  with a Paper-Clip Number Chain.                                                                                                                                                                               (animate this image)

First we use our practice to use multiple chain clips to create the unit 1. As 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 unit 1. In this example we select 10 clips to represent the number 1, so each paper clip represents 1/10. Now we can proceed and create a right-angle triangle, each of which having base and height of length 1 or consisting of 10 paper 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 than 14 clips, representing 1 and 4/10, or 1.4. This is our first estimate. If we used 100 paper clips to represent the number 1, the length of the hypotenuse will be more than 141 clips, representing 1 and 41/100, or 1.41, and so forth.

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.