Addition and Multiplication in Cutlists

Cut lists are an essential woodworking tool for calculating lumber needs.  Lumber is bought in large slabs at the lumber yard and then milled into smaller pieces that will comprise the finished piece of furniture.  Woodworkers use cut lists to calculate what size slabs they should buy at the lumber yard in order to ensure that there will be adequate supplies of wood to mill into furniture components back at their wood shops.

		Final Size	Rough Size
Part Name:	#	T x W x L	T x W x L	Board Footage
Legs	4	1.5” x 1.5” x 29.25”	2” x 2” x 30”	3.33 BF
Aprons	4	¾” x 3.5” x 12”	1” x 4” x 13”	1.44 BF
Top	1	¾” x 17” x 17”	1” x 21” x 18”	2.63 BF

(NCTM Standard: Number and Operations)

Leg Splay

The compressive strength of wood typically exceeds its tensile strength by a factor of 5.  Thus the strongest way to position wood is to have the grain aligned parallel to the force it is resisting.  To apply this principle in a chair design would result in the legs being entirely vertical.  While this design is theoretically strong, it is also very unstable.  As a compromise, chair legs are typically splayed out at 3-7 degrees in order to gain stability without compromising too much of the wood's compressive strength.

(NCTM Standard: Measurement)

Ergonomics

Human bodies come in a range of sizes.  When designing a custom piece of furniture for a unique client, a furniture maker can afford to customize all aspects of the design to the customers height and body dimensions.  When designing furniture for a restaurant or general use, furniture should suit the largest possible number of body types.  The study of the the average proportions of the human body is referred to as ergonomics.  By compiling extensive measurements of human bodies and processing those numbers into mean and average dimensions, the ergonomic studies that began in the 1960s have greatly increased the furniture designer's awareness and ability to support the human form.

                       

(NCTM Standard: Data Analysis and Probability)

Blade Duration

Factors such as dryness, density and silica content have a significant impact on the longevity of a sharp cutting edge in the tools being used upon different timbers.  Sharpening a plane blade requires a 5 minute interruption in work.  These 5 minute interruptions can accumulate to a significant loss of time in the course of a large project.  The choice to design in dense, silica rich timbers must be weighed against the added expense of additional sharpenings.  The chart below reflects the frequency of sharpening required by woods representing a range of densities.

            

Species	Density measured as Specific Gravity	Silica Content	Duration of a cutting edge
Pine	.350	None	45 minutes
Walnut	.63	None	15 minutes
Purpleheart	.750	High	2 minutes

(NCTM Standard: Data Analysis and Probability)

Time Analysis

Keeping track of the time needed for woodworking operations is an essential part of respecting customer deadlines as well as the profitability of the business.  By maintaining accurate records of how much time was required in past woodworking operations, woodworkers are capable of making accurate future estimates that ensure timely delivery and accurate pricing.

Task	Time needed	# of repetitions	Total time
Set up cross-cut saw	10 minutes	1	10 minutes
Cut lumber to length	15 seconds	20	5 minutes
Set up mortiser	30 minutes	1	30 minutes
Cut mortises	1 minute	20	20 minutes

(NCTM Standard: Measurement)

 

Analyzing Precision

The common knowledge in woodworking circles is that mortise and tenons should fit with a tolerance of 1/64”.  In Joe's experience, this tolerance is too great.  The most common type of wood glue, poly-vinyl adhesive (PVA), relies on the glued pieces of wood being in contact with each other or near enough to each other to allow for a polar attraction. The range of the attraction is less than .003”.  The glue itself has no spanning strength.  1/64” equates to .016”.  Thus if there is a 1/64” difference between the mortise and the tenon there can be up to .008” between either side of the tenon and the walls of the mortise.  While this amount of glue might seem to be insignificant, it makes a big difference in strength the finished product. Joe has seen glue fail in joints with tolerances over .01" but never in joints with tolerances of .005" or less.

(NCTM Standard: Measurement)

Working with Fractions

Most furniture is still measured in imperial measurements.  As a result, conversions between feet and inches are frequent and fractions with unlike denominators must frequently be added, subtracted, and multiplied.

(NCTM Standard: Number and Operations)

Spatial Reasoning

When designing a piece of furniture Joe needs to account for the comfort of the end user as well as the logistics of delivering the furniture in his vehicle and through the entrances in a building.  With larger furniture pieces, delivery can present a challenge in the area of three dimensional reasoning.  By designing this 9’ long bench to nest upon itself and fit inside his Subaru, Joe saved the customer several hundred dollars in delivery fees.

(NCTM Standard: Measurement)

Board Feet

Lumber is sold in a unit of volume known as board feet. A board foot is defined as 144 cubic inches of lumber.  A board foot can be comprised of a board that is 1” thick, 6” wide and 12” long.  Another example of a board foot is a board 3” thick, 8” wide and 6” long. 

(NCTM Standard: Measurement)

Layout

When laying out irregular-shaped furniture parts, furniture makers need to layout and trace their templates to maximize the number of parts that their lumber can yield.  Sometimes these patterns are regular, but often furniture makers need to exercise a creative grasp of geometry in order to maximize yield when avoiding knots and other defects in their lumber.

(NCTM Standard: Measurement)

Blade Angles in Hand Planes

A Hand Plane are a very simple tool that holds a blade at a fixed angle.  Woods respond differently to the planning angle and the angle at which the blade is beveled.

Typically speaking, soft woods are cut best by low planing angles (37-45 degrees) and low blade bevel angles (23-25 degrees).  Hardwoods are cut best by higher planing angles (45-60 degrees) and higher blade bevel angles (30-35 degrees).

The reason for this difference is twofold.  Low bevels produce a sharper but more fragile cutting edge.  Higher bevels produce a blunter, but more durable cutting edge.  A low bevel angle will slice softwood for a long time, but will rapidly chip when used on hardwood.

High planing angles have the effect of scraping the surface of the wood, while low planning angles have a slicing effect.  Soft woods tend to be crushed by the scraping effect, while hardwoods are more resilient and can produce a smooth finish when scraped by a high planning angle.

(NCTM Standard: Geometry)

Measuring and Fitting Tenons

The mortise and tenon is one of the oldest and most common methods of joining two pieces of wood at a perpendicular (or near perpendicular) angle.  A mortise is the hole in a piece of wood (think mortar), and a tenon is the piece of wood that lives in the mortise (think tenant).

A tenon must be smaller than the mortise it is entering, but the tenon must also fit snugly enough inside its mortise that the tenon will not pull back out after the glue dries.  I mill my tenons to be .003" smaller than their mortises because I have found this to be an ideal compromise between easy entrance and a snug fit.  Typically, I saw my tenons to be slightly larger than my mortises, and then I plane down the tenons removing .001" of wood per pass until the tenon is .003" smaller than the mortise and fits perfectly.

(NCTM Standard: Number and Operations)