Shadows Portfolio

The unit problem was to find a formula that expressed "s" (the length of a shadow) in terms of the variables L, D, and H (the unit problem is expressed in the diagram below).

Some of the components that we had to learn in order to solve the unit problem were similarity, congruency, right triangles, trigonometry, logical reasoning, and proofs. We began by examining shadows and the variables that affected the lengths of them; in homework 2 my group took the height of the light source and varied it to see if it made a change in the length of the shadow. When we did similarity we began by drawing simple objects and shrinking them or enlarging them to show the similarity. In homeworks six through thirteen we found how angles must be congruent and sides must be proportional in order for two triangles to be similar to each other. Proportions were another main idea in this unit because they gave us an understanding on how two ratios must relate to each other.

Although I had previously seen proportions homework ten gave me a clear visual on how they greatly relate to triangles since sides must be proportional for two triangles to be similar. For congruency I think I knew pretty well yet the supplemental worksheets helped me understand it more thoroughly. I found that congruency shortcuts helped me determine whether or not two triangles were congruent in a much faster and simpler way. Right triangles were mainly used in the supplemental worksheets and in the unit problem since side "d" is the horizon and sides H and L are completely vertical thus creating a right triangle. Also, when we did the activities with the mirror on the floor they showed right triangles since our bodies were vertical and the floor was the horizon.

Another main homework that used right triangle was homework nineteen because in order to find the height of the tree I drew two right triangles and made a proportion to figure out side L which was the height of the tree. Logical reasoning and proofs helped me to prove the similarity between certain angles or congruency between any lines. The reasoning gave me things to back my information up with and showed me that by building up on the information I know I can prove things to help me get to a solution. Trigonometry gave me a completely new way of finding new solutions to missing information in any kind of triangle. Cosine, Tangent, and Sine were a completely new concept to me but once I understood them I was able to find new information and it made the concept of missing information much easier for me.

In my opinion homework 23 was the assignment that gave me the best understanding of cosine, tangent, and sine because it showed me how to use them to find missing information. One last main idea that I think was beneficial to solving the unit problem was finding angles when I knew the measurement of other angles; the supplemental worksheets helped me with this greatly and knowing the geometry terms and what they meant also helped me to define these angles. All of these components gave me the knowledge to solve page 64 which was the solution to the unit problem H(S+D) =LS.

The object of the N by N window was to create a formula that would give me the total length of wood strip needed to build square windows of different sizes. I was given a simple 3 by 3 window and I needed to find different sizes of windows to eventually find a pattern that would help me find the formula. One key idea included is the use of logic and reasoning and the use of an in-out table. I think the main key idea that helped me to solve this problem was logical reasoning and finding a pattern within the data I recorded once I made more windows. This piece is included and relates to the unit problem because although the setting is completely different from the unit problem it still relates to finding a general formula.

In the Statue of Liberty assignment we needed to find the length of lady liberty's arm and two other body parts of our choosing. We were given the measurement for the nose of lady liberty. One main idea used in this activity is the use of proportions because we needed to measure a classmate's nose to find the ratio and find two other measurements on the Statue of Liberty. We measured a classmate's arm and leg and used the ratio we found to make a proportion and find the measurements for Lady Liberty's arm and leg. This relates back to the unit problem because it shows similarity by using ratios just like the two triangles are similar in the unit problem.

I chose homework 15 as my assignment on similarity because it showed me that triangles within triangles can be similar as shown in the unit problem. The object of this homework was to find when a smaller triangle created inside of a larger one comes out to be similar. We had to come up with examples that showed when and how triangles within triangles would come out similar to each other. This related back to the unit problem because as I stated before it shows similarity when a triangle is in another one as shown in the unit problem. This homework helped me in the development of solving the unit because it gave me a better understanding of the similar triangles.

When I first read the small section on trigonometry I was completely confused and I didn't understand anything regarding sine, cosine, or tangent. After reading it a couple of times and help from a classmate I understood the concept of them and I was able to comprehend what homework 23 was asking me to do. This homework gave me a chance to apply sine and tangent for the first time and since the problems were not so complex I was able to clearly understand what was being asked of me. I needed to find the sine and tangent for part one of the assignment and I needed to find the tangent for part two of the assignment. The mathematical key idea in this assignment was trigonometry and the use of sine, cosine, and tangent; this relates back to the unit problem because it can help us find a missing side or angle.

The exterior angle and polygon sums activity asked us to find the relationship between an interior and an exterior angle, what the sum of an interior and an exterior angle, the sum of all the interior angles in a polygon, and the total number of degrees in all of the exterior angles. The main mathematical concept of this assignment was to relate angles to each other. This relates to the unit problem because it gives us a better understanding of how the angles of a polygon are formed and what they are all equal to.

The last homework assignment that helped me develop key ideas to solve the unit problem was homework 19. I was asked to use a proportion to find the height of a tree using similar triangles. This related back to the unit problem because the diagram I drew looked almost identical to the one we used to show the unit problem.