Activity 3: Fashion Design: Patterns and Weaving (Grade Level 811)
The Geometry of Bicycle Designs 
Using Symmetry to Create Corporate Logos 
Patterns and Weaving  Career Connections 
More Math Concepts
Standards:
Standard 2: Patterns, Functions, and Algebra (Deriving a Model and Applying the Quadratic Formula)
* 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 knowhow. 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 N^{2}
– 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.
