A Craftsman Divides a Rectangle

Objective: To demonstrate how a craftsman such as a carpenter, cabinet maker, or sheet metal worker can divide a rectangular piece of material into a number of regions of equal size using a pencil and straight edge. The technique illustrated in this demo avoids cumbersome and sometimes inconvenient direct measurements.

Level: This demo is appropriate for a geometry class, an introductory modeling class, or possibly a general mathematics class for industrial arts.

Prerequisites: Students should be familiar with basic geometry involving congruent triangles, angles made when parallel lines are cut by a transversal, and simple measurements.

Platform: No electronic technology required.

Materials Required: Rectangular pieces of paper, cardboard, or wood;  ruler, yard or meter stick, (carpenter's ) measure; pencil or pen.

Instructor's Notes:

Background: When a craftsman needs to divide a rectangular piece of material into a number of equal portions they of course can proceed using direct measurements. In many of cases the material may be of an odd size and dividing its length into n equal segments  may involve fractions of inches or centimeters that are difficult to locate precisely. For example, a cabinet maker needs to place handles on a drawers as illustrated in Figure 1.

 Figure1a. Figure 1b.

In Figure 1a the drawer face appears to be have been divided into quarters and the handles centered at the 1/4 and 3/4 positions. In Figure 1b the drawer faces of the chest appear to have been divided into sixths and the handles centered at the 1/6 and 5/6 positions.

Another application involves the construction of duct work for an air conditioning system. See Figure 2. If a square duct is required then a long piece of sheet metal must be divided into quarters and bent along the quarter marks to form the duct.

 Figure 2.

Discussion: We will first illustrate the "equal division" technique used by various craftsman using a particular problem that can be used in the classroom and then provide a general overview.

Example: An 8.5" by 11" standard sheet of paper is to be divided into three equal regions as shown in Figure 3.

 Figure 3.

Each of the regions is to be 8" by 11/3". Since most common measures are given to 1/16 of an inch, it requires some estimation to place marks at 11/3 and 22/3 of an inch at several place and then connect them with vertical lines as shown in Figure 3.

An alternative procedure is to take a standard 12" ruler and do as follows:

• Since we want three equal regions, we note that the 12" ruler has three equal segments of 4" each.
• Place the left end of the ruler at the left edge of the sheet of paper.
• Move the right end of the ruler in a diagonal position across the paper so that the right end is at the right edge of the sheet of paper.
• With a marker, draw the diagonal along the edge of the ruler connecting the two edges of the paper.
• With your marker put a dot on the paper at the position of the 4" location along the diagonally positioned ruler. Make another mark at the 8" location.
• Move the ruler parallel to itself to draw a second diagonal as described above.
• With your marker put a dot on the paper at the position of the 4" location along this second diagonally positioned ruler. Make a another mark at the 8" location.
• Position your ruler to align the pair of 4" marks you made and draw a line through them. Do the same for the 8" marks.
• You now have the sheet of paper equally divided into 3 regions as shown in Figure 3.

These steps are shown in Figures 4a - e.

 Figure 4a. Figure 4b. Figure 4c. Figure 4d. Figure 4e.

The frames of Figure 4 can be seen as a Quicktime movie  by clicking here.

Why does the procedure work? The answer is, some easily constructed right triangles are congruent. Figure 5 shows three triangles that can be shown to be congruent and so "equal division" follows. A key step is to recall that each of the green segments have the same length, in this case 4". Each triangle is a right triangle and the acute angles at the right-hand vertices of each triangle are shown to be of equal measure using properties parallel lines cut by a transversal, namely the diagonal. This is a nice application for a geometry course.

 Figure 5.

To see an annotated construction of the "equal division" of the sheet of paper in a Quicktime file click here.

General Case: It is easier to make marks at locations that appear on your ruler or measure than to estimate a position not explicitly appearing on it. For instance, the 11/3 and 22/3 inch positions on a ruler are not on a ruler so if we were to measure horizontally across the sheet of paper of the example we would need to estimate these positions. With this in mind we try to arrange to use our measuring device as follows.

For a rectangular piece of material of width w that it is to be "equally divided" into n regions we use a portion of the measure which consists of n segments of equal length that occur at marks on the measure. The left end of the measure is placed at the left side of the material and we stretch it diagonally to the right side of the material.

For a 30" by 30" poster board that is to be divided equally into seven regions we could use a portion of a carpenter's rule or yard stick that is 35" long. Position it diagonally as discussed in this demo and mark at 5" intervals along the diagonal.

For a 4' by 8' sheet of plywood that is to be equally divided into six 8' strips we could use a 6' portion of a carpenter's measure. Position it diagonally and mark at 1' intervals.

If large construction materials it is often easy to use a portion of a measure which consists of n segments of equal length.

In the classroom: For in class demos an 8.5" by 11" sheet of paper works nicely if you want to equally divide into six regions. A 3" by 5" card works well if you want to equally divide it into four strips each 1" long.

Reference:

Measuring, Marking & Layout, A Builder's Guide, by John Carroll, The Taunton Press, Inc. 1998, p. 29 has a discussion of 'Dividing a Distance into Equal Segments'. The technique illustrated in this demo is referred to as the slant-rule trick.

Credits:  This demo was submitted by Sean Comfort, an undergraduate mathematics major at Temple University, who, in a continuing life, is also a skilled carpenter. The animations, figures, and accompanying text were constructed by David R. Hill, Temple University. This demo is included in Demos with Positive Impact with their permission.

DRH 9/29/2004     last updated 5/18/2006 DRH

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