Chapter 1 Project

Patterns, Patterns Everywhere!

An activity to demonstrate the use of mathematical expressions to represent patterns.

Mathematics can be found in nature in the form of patterns. These patterns appear in the symmetry of snowflakes, the grouping of petals on a flower, and even the radial symmetry of a drop of water splashing into a pond. Mathematics can be used to explain and analyze these natural phenomena.

1. Consider the following pattern.

   

   the first iteration is a single dot centered in the image. The second iteration is 4 new dots surrounding the original 1 dot, one at each of the four corners of a box. This gives a total of 5 dots. The third iteration adds an additional 4 dots to the second iteration's 5 dots. This gives a total of 9 dots. The new 4 dots are placed at each of the corners of a box and now it resembles an x shape.

   1. Describe the pattern in words.

   2. Describe the pattern using a sum of numbers for each iteration.

   3. Fill in the following table using the left side as the iteration of the pattern and the right side to represent how many dots are in that iteration.

      Iteration	Number of Dots
      $1$	$1$
      $2$	$5$
      $3$	(blank)
      $4$	(blank)
      $5$	(blank)
      $6$	(blank)
      $7$	(blank)
      $8$	(blank)
      $9$	(blank)
      $10$	(blank)

   4. Create an algebraic expression that represents this pattern using n to represent the number of times the pattern has been drawn.

   5. How many dots would be in the 100^th iteration?

Now we'll look at the pattern formed by the shell of a chambered nautilus, which is a mollusk that lives in the South Pacific Ocean. Due to its distinctive shape, the nautilus shell has been used in art for centuries. The first image shows a cross section of a nautilus shell with a clear view of its chambers. The second image illustrates how the shape of the shell can be approximated by a series of squares that increase in size. This increase in size is a famous pattern often found in nature, the Fibonacci sequence.

A diagram of how the shape of the nautilus shell can be approximated by a series of squares that increase in size moving in a counterclockwise direction. It starts with the smallest inner spiral as a 1 unit by 1 unit square. Another 1 unit by 1 unit square is added on top of it. To the left is added a 2 unit by 2 unit square. Below that is added a 3 unit by 3 unit square. To the right is added a 5 unit by 5 unit square. Above that is added a 8 unit by 8 unit square. To the left is added 13 unit by 13 unit square that completes the spiral.

2. Analyze the pattern of the nautilus shell and express the pattern with an algebraic expression. From the pictures of the nautilus shell, you can see a pattern emerge. The inside of the shell starts with a small single square that is one unit per side. Then larger and larger squares are added in a counterclockwise direction using the following number sequence. The side lengths of the squares are described by the pattern $1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,\dots$

   1. Describe the pattern in words.

   2. Fill in the following table using the left side as the iteration of the pattern and the right to represent the side length of each square.

      Iteration	Number of Dots
      $1$	$1$
      $2$	$1$
      $3$	(blank)
      $4$	(blank)
      $5$	(blank)
      $6$	(blank)
      $7$	(blank)
      $8$	(blank)
      $9$	(blank)
      $10$	(blank)

   3. Create an algebraic expression that represents this pattern using n to represent the number of times the pattern has been drawn.

3. Create your own pattern. Have a classmate describe and create an algebraic expression for the pattern you created.