Activity 3: Fashion Design: Patterns and Weaving (Grade Level 8-11)

The Geometry of Bicycle Designs | Using Symmetry to Create Corporate Logos | Patterns and Weaving | Career Connections | More Math Concepts

Standards:

* These standards have been adopted and are based on the information from Principals and Standards for School Mathematics: Discussion Draft, October l998, National Council of Teachers of Mathematics.

Objectives:

• Use quadratic equations to solve problems in weaving.
• Determine numerical patterns to find a formula

Activity 3

Shirts and sweaters with interesting and attractive designs can be made by weaving threads of different colors. While the choice of patterns and colors depends on the designer’s creativity and good taste, the actual execution of the weaving depends on mathematical know-how. Believe it or not, the quadratic formula actually comes into play.

Suppose a designer wants to produce a sweater with a fairly simple color pattern: a block all one color except the last thread, each time followed by a block with one more thread of a new color, and continuing until the last block is made up one thread from the first color and all the rest from the new color.

For example, suppose the first block is all purple and the one new thread is green. Keep adding one more green thread until the entire block is green except for the first purple thread. The resulting pattern is a gradual change from all purple to all green:

Weaving machines are made up of parts called “warps” and each warp is divided into “blocks.” Warps have a fixed number of threads (say, 300). To produce the desired color blend pattern, a weaver needs to divide the warp into equal sized blocks: the first all purple with one green thread, the next all purple but with two green threads, followed by all purple with three green threads, and so on, until the last block is one purple thread followed by all green threads. How does the weaver figure out the block size and the number of blocks given the 300 thread warp?

Like so many things in mathematics, start with something simple and build up to a more complicated design. Consider an example of a 72 thread warp. Notice that it is made up of 8 blocks, and each block has 9 threads (8 times 9 equals 72).

Below is a warp made of 72 threads in which each block has 9 threads:

The numbers (threads per block and blocks per warp) work out well in this example, but in the real world of weaving, the numbers are not this easy. The warp size is usually given; it is not adjustable to the design pattern. For example, how many blocks and how many threads per block will be needed for our color blend pattern on a warp of 300 threads? Look for a mathematical pattern.

Start with two threads: one purple thread, P, and one green, G. If you add a thread, you can now start the blend: two purple and one green (PPG), followed by one purple and two green (PGG). Make a table to keep track of the block size (number of threads used in the block), the number of blocks, and the warp size (total number of threads).

Notice what happens when you add a fourth , fifth, and sixth thread to the block.

 Color Pattern Block Size # of thread Number of Blocks Warp size Total # of threads PG 2 1 2 PPG PGG 3 2 6 PPPG PPGG PGGG 4 3 12 PPPPG PPPGG PPGGG PGGG 5 4 20 PPPPPG PPPPGG PPPGGG PPGGGG PGGGGG 6 5 30
Do you see a pattern emerging? If the block has N threads, there will be N – 1 blocks, and the warp size will be N(N – 1).

Now you have what you need to answer the question of how many blocks and what block size to use for a 300 thread block. You need to find N such that
N(N – 1) <300, and N must be a whole number. So N(N – 1) is close to or equal to 300.

You can solve N2 – N – 300 = 0 using the quadratic formula:

So, the largest whole number less than or equal to N is 17. If you make a block of 17 threads, there will be N – 1 or 16 blocks for a total of 272 threads. You’re 28 threads short of 300. Split the difference: put 14 purple threads at the top of the warp and 14 green at the bottom to bring the total to 300 (14 + 272 + 14).

1. Using the same color blend pattern as in the example, suppose the warp size is 500.

a. What is the number of threads per block? How will you handle the excess thread needed to reach 500?

b. Again with the same pattern, suppose the warp size is 800. What is the number of threads per block? How will you handle the excess thread needed to reach 800?

2. Imagine a design that involves two colors (purple and green), but use this pattern: start with one color and keep changing the top and bottom thread to the second color until all threads are the new color. Here is an example:

Determine the mathematical pattern for the number of threads per block, the number of blocks, and the number of threads per block for a 300 thread warp.

a. Complete the table below to find the mathematical pattern.

 Color Pattern Block Size # of thread Number of Blocks Warp size Total # of threads PP GG 2 2 4 PPP GPG GGG 3 3 PPPP GPPG GGGG PPPPP GPPPG _____ _____ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______ ______

b. If N represents the number of threads per block, find an expression that represents the number of blocks.

c. Find an expression to represent the total number of threads.

d. Find the whole number value for N that will produce a warp of 300 threads.

e. If necessary, add threads to achieve the 300 thread warp. Describe where each thread will be placed.