Shadows Portfolio Imp 2

Over the duration of six weeks, we worked in a math unit called "Shadows." During the unit, we worked to solve the unit problem, which was to find a formula for how long a shadow was. When first given the unit problem, I considered it a simple task but soon after I realized there were many steps and ideas that needed to be learned before the unit goal could be reached. Throughout these six weeks, we learned about trigonometry, similarity, patterns, congruency, and using angles to solve problems. These new math ideas were just things we needed to know to find out our bigger goal for the unit.

One of the first activities we did using manipulatives was physically measuring a shadow or Shadow Data Gathering. When measuring a shadow you need to use four variables, L for the height of the light source, D for the distance from the object to the light source, H for the height of the object and S for the shadow length. These four variables would be the ones you use in the formula for how long a shadow is, our unit goal. This activity helped develop my understanding throughout the unit because it was a hands on experience of measuring a shadow, which relates you back to the unit goal. This also helped because instead of just seeing it in pictures or solving it in homework's such as the sun problem and the lamp problem we got a visual, hands on interpretation,

After measuring shadows and finding out what variables were needed to solve the unit problem the next thing we learned was how to create formulas using an In-Out table. In POW 17 making an In-Out table and identifying patterns was key to finding the formula needed and it is the same situation for solving our unit problem. Although formulas do not relate to shadows, they relate to what our unit goal is, which is finding a formula.

After using In-Out tables, finding formulas and measuring shadows we took a turn in our studies to a more geometric side of this problem. The next concept we studied was similarity. Homework 6 was a good introduction to this new key idea because we saw similar shapes and why shapes are similar to one another. It also gave a segway to another concept in the unit we needed to know. This concept is how angles are important in finding similarity in between shapes. Shapes that are similar are proportional to one another but one is bigger or smaller than the other. They have the same angle measurements and if they do not the angle measurements are proportional to each other. Although it may not seem like it similarity has a lot to do with shadows; it actually does. To see the idea illustrated further, look at the picture below.

The picture creates two triangles, one between the person and the shadow and one between the tree, the person and the shadow. These two triangles are similar.

Since triangles are the shapes that are used in the shadow unit problem (see figure one) we took a closer look at them. Congruency is a word relevant to triangles because when two triangles are congruent, they are the same. In addition, when we took a closer look at triangles we studied interior, exterior, and right angles. A right angle is equal to 90 degrees. Knowing these new angle terminologies, it helped us understand how to find the degrees of an angle, which would come in help later with the unit when given various worksheets on the topic.

While studying similarity, we also took a closer look at how to set up proportions to see if triangles were similar or not. Homework 16 was a good example of setting up proportions because it showed you how to find proportions inside and outside of the triangle. Overall, proportions are a key part in the shadow unit because you use them not only to find similarity but in certain homework's you used them to find heights.

After learning about all the details of similarity, congruency and proportions we studied angles of approach and departure. An angle of approach is the angle between the mirror and incoming ray of light from the flashlight. The angle between the mirror and the ray bouncing off the mirror is called the angle of departure (see picture for further understanding).

This concept was used in Homework 17 and many other assignments. Although I am not quite sure how learning about angles of approach and departure helped us reach our unit goal of finding the formula, it was useful to know for solving other equations we had to answer on our road to answering the unit goal.

Many times when trying to figure out angles we had to use other information we had such as when we were working on the Shadows Supplemental Worksheet or when trying to figure out a transversal. A transversal looks like this (see picture). When trying to figure out the angle measurements of a transversal you have to compare alternate exterior and interior angles. You also know that a and b have to add