Cover Letter
The Orchard Hideout is a themed unit about circles that comes from the Interactive Mathematics Program (IMP). The central problem of the unit is about two people who have planted trees in a circular pattern. The activities in the Orchard Hideout unit help students work toward a solution to the central problem by applying knowledge about circles and coordinate geometry. A key part of this unit involves developing formulas for the circumference and area of circles. For me, a big idea I take from this unit is that a rich problem will promote the use of different mathematical abilities.
This portfolio contains my work on selected problems from the Orchard Hideout unit. The pieces I have chosen highlight the work I found most meaningful and does not include everything I worked on during the unit.
This portfolio contains my work on selected problems from the Orchard Hideout unit. The pieces I have chosen highlight the work I found most meaningful and does not include everything I worked on during the unit.
Selected Problems
One of the tasks I found meaningful was “More Mini Orchards.” The goal is to find how big the radius of the trunk of each tree needs to be for the circular orchard to become a hideout. We are asked to find this out for an orchard of radius 2 and radius 3. A key point is that the radius of each tree trunk depends on the radius of the orchard itself. I found that thee orchard of radius 2 has 12 trees in it. I sketched this out in my Interactive Notebook where each tree can be was represented by a point on the coordinate. Rather, I found the radius of each tree trunk would need to be ½ in order for the trees to touch. I used the Pythagorean Theorem to prove that the radius of each tree trunk had to be ½. But, when I worked on this problem I did not take into account the lines of sight. Accounting for lines of sight would have helped me see that the radius of the tree trunks could be a little less than ½. This problem was the first homework problem I did for the Tree Orchard unit. One take-away from this problem was that there were many ways I could have approached the problem. This helped me see that there are many mathematical abilities needed to work on a quality problem such as this one. For example, in this problem I drew a picture to help me visualize the problem. Also my knowledge of circles and triangles helped me find the radius of the trees.
“Equally Wet” was another task that helped me grow this semester. In this problem, two flowers are planted in a garden and we need to find where to put a sprinkler so that the flowers receive equal amount of water. I began by placing the two flowers 4 units apart on a coordinate grid in my Interactive Journal. The obvious choice for the sprinkler was at the midpoint of these two points. But I wanted to know if there were other places I could put the sprinkler. So I drew a straight line along the midpoint and placed the sprinkler there, then I used the Pythagorean Theorem to prove this point was equidistant from the flowers. The next part asked me to find where to place the sprinkler if there were 3 and 4 flowers. I discovered there are conditions for when the sprinkler can get the flowers equally wet. In short, if the distance between the flowers were equally spaced apart, then I could place a sprinkler at the intersection of the perpendicular bisectors. This POW also helped me realize the importance of drawing a diagram in order to visualize the problem. Sketching it out was in fact necessary for me to solve it.
Another task that I found meaningful was “Big Earth, Little Earth.” The problem wants us to find out how long of a piece of string wrapped around the equator of the Earth would be. At the same time you wrap a piece of string around the equator of a globe and need to know how long it would be. Then, if we extended each piece of string by 1 foot. Now, how much bigger would each radius have to get? The first step I took to solve the problem was to find the circumferences of the globe and the Earth. This told me how long each piece of string needed to be. I converted each measurement into feet to make the next part simpler. Then, I added 1 foot to each length and found the new radius using the formula r=C/2. The final step I took was to find the difference between the new radii and the original radii. I found that for both the globe and the Earth, the change in radius came out to be 0.16 feet. One reason I chose to incorporate this task is because I really like doing computations and following formulas. So in that sense, this problem was fun for me to work on.
Statement of Personal Growth
I have definitely grown as a teacher over the past semester. I have began to feel more comfortable in front of the class and I feel more confident in my abilities. I have learned so much from both of my clinical practice experiences. This semester I have fully taken over the class and began to experience it takes to be a teacher. I would say I have began to ask more thoughtful questions. But I still have some trouble with classroom management. But the more confident I become the better my classroom management becomes.
The Orchard Hideout unit has helped me develop a deeper understanding of the geometric concepts. The unit never gave procedures to follow. And like the Baker’s Choice, it guided me through the unit by asking questions like “why” instead of “what”. The Orchard Hideout Unit emphasized the importance of mathematical discourse. The discussions we had this semester challenged me helped me gain a deeper understanding of geometric concepts. Reading Designing Groupwork by Cohen and Lotan also helped me identify strategies for implementing productive groupwork. I intend to incorporate these collaboration strategies into my future teaching practice. Overall, I enjoyed this unit because it challenged my need to follow procedures and more importantly, it made me think deeper about mathematics.
The Orchard Hideout unit has helped me develop a deeper understanding of the geometric concepts. The unit never gave procedures to follow. And like the Baker’s Choice, it guided me through the unit by asking questions like “why” instead of “what”. The Orchard Hideout Unit emphasized the importance of mathematical discourse. The discussions we had this semester challenged me helped me gain a deeper understanding of geometric concepts. Reading Designing Groupwork by Cohen and Lotan also helped me identify strategies for implementing productive groupwork. I intend to incorporate these collaboration strategies into my future teaching practice. Overall, I enjoyed this unit because it challenged my need to follow procedures and more importantly, it made me think deeper about mathematics.