Statement of the Mathematical Exploration
Through investigating the following pattern of tiles, we can explore the pattern in graphical form using a graphing calculator, and develop several algebraic rules to describe the pattern. Each of these algebraic rules can be shown to be equivalent expressions and related to the geometric arrangement in the pattern.

My Mathematical Exploration
Looking at these tiles above, I was able to continue the pattern for the next three terms. I created a table of values that showed the term verses the number of tiles.
 

Then I  inserted this data into my STAT lists on my graphing calculator (TI-83) and created a scatter plot.

From the data and the graph, I can tell that the pattern is not growing linearly. I hypothesized that the equation would be a quadratic and then analyzed the pattern and table to see if I could develop an algebraic rule. I generated two algebraic rules that would determine the number of tiles needed for the nth term. The two rules I used for the number needed in the nth term were  and . I graphed these equations with my Y= tool. I could see then that my rules corresponded with the data.

The nth term is represented by the variable X. I came up with the first equation by looking at the pattern. I noticed the area of the rectangle formed by the center tiles is equivalent to one less than the term multiplied by one more than the term (length times width). Then, there are two outside blocks, so I added two. The first set of tiles had no center tiles; therefore, the number of tiles is two: . This equation corresponds well with the geometric arrangement of the tiles. We take the area of the center rectangle and add two. From the tile arrangement, it is very easy to see how I put together my equation.

I formulated the second equation from the chart I made (term vs. # of tiles). I observed for each term that the number of tiles are equal to the term squared plus one. The second equation does not correspond well with the geometric arrangement as it is presented. However, if we were to rearrange the tiles so that the top row of tiles were turned and placed on the side of the center tiles, we could see a perfect square with side equal to the term number plus one extra tile. Below, I have shown that the two equations are equivalent.

Reflection on How the Technology Tool Helped My Conceptual Understanding

I had never looked at functions from a patterning perspective before this exploration. The visual pattern of the tiles makes the relationships between the terms of the function, along with their powers or coefficients, seem less abstract than just looking at a function by itself outside of any context.  It may be easier for some students to develop function rules from the tiles, than from the table of numbers.  For me, the tiles helped the table of numbers make sense.  Generated two different algebraic rules to describe the pattern challenged me to look at the problem in more than way. As I explained above, I was able to generate one rule based on the rectangular array in the pattern. This connection to my understanding of area was helpful. Looking for a rule based on the table made me think about the numerical pattern. From studying the list, I recognized that every number in the "number of tiles" list was one more than a perfect square number. This was a different type of pattern recognition than I did with studying the geometrical tile pattern. I feel like this exploration helped me see how patterns can be used to develop algebraic thinking in students and help them develop an understanding of a function. The ability to see patterns is not only useful in solving mathematics-based problems, but also other types of problems as well.
 

An Outline of a Four-Point Lesson Plan for Investigating Patterns With a Graphing Calculator

Objective: 8th grade students will analyze a given geometric pattern, generate a table of values and graph of their data, create at least two algebraic rules to describe the pattern, and justify why their rules are equivalent and are connected to the geometric pattern.

Phase 1: Problem Posing
    I will displays several geometric patterns on the board or overhead. I will challenge students to continue the pattern for the next several terms and discuss how the pattern is growing. They are to determine how many objects would be needed for the 100th term in each of the patterns and to eventually describe a rule for finding the number of objects needed at any term (n) in the pattern. Some sample patterns include the following:
 

1	2
3	4

Phase 2: Small-Group Investigation
The students will work in small groups of 3-4 to investigate a pattern. Students will have tiles, paper and pencil, and their graphing calculators. Each group will work on a pattern, with only two groups working on the same pattern. I will circulate and ask the students guiding questions concerning how they are modelling the pattern. During the group work, I will ask guiding questions that encourage the students to use the table of numbers, the geometric pattern and a graph of their data to help them describe how the pattern is growing. If the students are having difficulty writing a general rule given the nth term, I will focus them on the geometric pattern and ask them to describe the dimensions of the figures in terms of n. I will certainly encourage students to come up with different ways of describing their pattern and to make connections between the symbolic, geometric, and numerical representations.

Phase 3: Whole-Class Discussion of Investigation
For each pattern explored, I will draw a group's name at random and ask that group to present their exploration to the whole class. The other group that explored that same pattern will have the opportunity to compare and contrast their thinking with the first group's process.  All equivalent expressions for a pattern's rule will be written on the board to be explored at the end of the period.

Each pattern will be explored and discussed and I will guide the class in encouraging them to make the connections betwen the geometric, numerical, and symbolic representations. Each pattern will be graphed using the Stats Plot feature on the GC and the class will compare and contrast how each of the patterns are growing (1 & 2 are linear, 3 & 4 are quadratic).
 

Phase 4: Summarizing and Extending
After each group has presented their findings and justified their reasoning, I will summarize the findings for each pattern, especially the various symbolic rules. Students will record this information on a worksheet containing each of the four patterns.  For homework, the students are to further analyze each of the patterns and describe the similarities between 1 and 2, and then between 3 and 4. Students will then describe the differences between the pair of 1 & 2 and the pair 3 & 4. The students are also challenged to explain why the different symbolic expressions are equivalent. Although we have not done any formal simplification of algebraic expressions, I am hoping this task will lead us into this discussion tomorrow.