I. Resources: For our project, we got ideas from Ms. Sheehy in the Math Ed department. She gave us the following ideas: 1. Using a tape measure, drop a ball from a given height and measure the height of its first bounce. Change the drop height and measure the first bounce. Keep repeating the process. (linear) 2. Given a length of string, make a rectangular pen with the largest area. (quadratic)
II. Introduction: 1. Mathematical topic: The mathematical topic that we are planning on teaching, is to utilize some applications of linear and quadratic equations and show how these would apply in the real world, using handson investigations. 2. Where our topic fits into the curriculum: Our topic is extremely fundamental in the curriculum, because linear and quadratic equations are probably used in the most common realworld applications of mathematics. They are also fundamental in understanding all mathematics they would learn in the future, in any other math classes they may have in high school or in college. 3. Important information about our topic: On the topic of linear and quadratic equations, we believe that the concepts of slope, graph identification, and that the relationships between the variables are important to understand, as well as knowing that there are different ways to solve an equation besides algebraically. 4. Concepts that are difficult to understand related to our topic: We believe that the concepts most difficult to understand related to our topic are slope and possibly the relationship between the actual activities and the mathematics behind them that the students learn in class. III. Class Level: We are going to use our lesson is in a 9^{th} grade Algebra I class of 25 students, and will allow each one to be engaged in the learning process with the two handson experiments mentioned above. We also hope to incorporate technology, allowing each student to participate with Excel, along with a demonstration that we hope to do with GSP for the quadratic lesson and the rectangular pen.
IV. Activity: The purpose of our lesson is to help students understand that there are multiple ways of looking at the same concept or idea they may learn in class. Hopefully they can accomplish this by exploring situations and performing investigations that involve functions, one of the most fundamental and integrated concepts in all of mathematics. We hope that they will be able to see that there is a real connection between content and context, and understand that there are actually important relationships between the math they learn in the classroom and realworld applications. Of course we won’t know how much time the activities will take the class to perform, and therefore, we have planned two activities that we believe may take awhile just in case. Investigation #1: For the actual activities / lessons that we will do with the students, we will begin by introducing ourselves and then dividing up the students into 5 groups of 5 for our first experiment, investigating linear equations. We will divide the students up by having them each pick a Starburst out of a bag, which we will make sure ahead of time has 5 of each of the 5 colors, and then the students will match their color with others in the classroom. Then, we will hand out the data collection sheets, one for each group, and the instruction sheets, one for each student, and split ourselves up so each one of us will go with a group to perform the experiment. Then, the students will do the actual investigation and get the averages of their 3 trials for the five different heights they choose. (see data and instruction sheet) After the students have completed the experiment, one student from each group will write their data up on the board. Eventually, we should have 25 pieces of data on the board, 5 from each group. At this point, we would like to hold a discussion about the student's findings, and ask some questions. For instance:
1. What did you believe that the relationship was between the height that the ball was dropped and the height of the bounce?
Then we will have the students use Microsoft Excel to enter in their collected data and eventually make a graph with a scatter plot. Hopefully, the students will each be able to use a computer to graph on their own data, but if not, we hope to at least have the students be able to work in pairs. Again, we will discuss their findings and ask them questions about what the graph means and what the line is telling us. We are going to discuss making predictions with the data using the line and discuss the implications of the slope. Also, we will show them how to add a trend line in Excel, along with an equation for their graph. After we discuss their graphs and the data, we are going to bring up the real world applications that are related to this type of experiment, such as manufacturing tennis balls, basketballs, bowling balls, ping pong balls, etc. as well as professional athletes and how their performance is affected by the way that a piece of sports equipment is manufactured. Finally, we can ask the students what they believe would happen if we performed this experiment with a bowling ball or a ping pong ball, and what the graph (including the slope) may look like. Investigation #2: For this investigation, we are going to try and have the students work individually. We will begin by having each student sit at their desks, or tables, and give them each a piece of string 100cm. long, three pieces of masking tape, ruler, as well as the instruction sheet and data collection sheet for the experiment. We will start with a discussion about area, the formula for the area of a rectangle and why it works. We will also discuss how to record their trials in the data sheet. The students will have two trials with choosing a side length to construct a rectangular pen with the biggest area. They will use the edge of the desk or table as a side for the pen and measure their side length (the side perpendicular to the edge they are using). Therefore, we will also have a brief discussion on the concept of perpendicular if they are unfamiliar with it. Then, the students will perform the experiment, and then on the second trial, attempt to change the side length, x, in order to receive a greater area for their pen. After their two trials are recorded, we will start on one side of the room and have the students quickly call out the length of their side, x, and then the area they came up with for the two trials. We will have the students record this data on their sheets, and at the same time, one of us will hopefully be able to use an overhead computer with Excel to punch the data in also so the whole class can see. Before we graph it in front of the class, we will again have a discussion about what the data means and see what they think the graph may look like. We will pose similar questions to the ones stated earlier in investigation #1. Afterward, we will ask the students if they remember how to use the graph tool on Excel and have them lead us (if possible) through creating a graph for our data points. We will then discuss the graph and explain that it represents a quadratic function. We can ask the students why it is quadratic, and discuss that the graph has a maximum point as well. Also, we can ask what the maximum point on the graph means and see if any of the students came close to the maximum area in their experiment. Also, if time, we can discuss the limits of the graph, and explain that the largest the side x could be is 50 cm., where the area would be equal to 0. Finally, we will wrap up the activity by tying the investigation into a real world situation of building a fence or building a pool in your yard. We can also discuss how important the concept of area is, and that architects, engineers, contractors, construction workers, homeowners, farmers, etc. all use area in their everyday jobs. Depending on how much time we have left, if any, we will tie in the two projects together and discuss how there is many different ways to see and learn a particular concept, such as solving a problem algebraically or by looking at a graph of spread sheet in Excel, and that all concepts in math have realworld applications to some extent. We also will hopefully have a minute or two left to be able to do a demonstration on GSP where we would show how the largest area problem works, where the length of the side, x, would be a dynamic vector, so that we could adjust the length and show them how the measure of the area changes.
V. Assessment: We plan on assessing the students by asking lots of questions about the experiments and about the results that they come up with as a class. As an incentive, we are going to bring Starbursts and Hershey Kisses and hand them out as the students attempt to answer our questions. At other points during our discussion, we will have students enter data into Excel, which we will explain how to do beforehand, and can assess them by wandering around and looking at what they've done on the program.

