Representations of: Mathematical Combination and Pascal's Triangle

In my last action project for prove it or lose it, we were looking at combinations.  This action project is for my course "Prove it or Lose It.” In class, we have gone over permutations, combinations, and Pascal's Triangle. We saw how pascal’s triangle it symmetric and how it shows reflections. We also watched videos and saw slideshows on permutations and Pascal’s Triangle. This unit has made me think about how math does relate to things that we wouldn't expect.

A mathematical combination is a selection of things from a group. The order of those things doesn’t matter in combinations, unlike permutations. FOR EXAMPLE: There are 10 marbles in a jar and each marble has a number from 1 to 10 on it. Three marbles are selected at random. How many different ways are there of selecting the three balls?
Formula: (n/r) = n C r. = n!/r! (n-r)!
Answer:
10C3 =10!=10 × 9 × 8= 120
             3! (10 – 3)!3 × 2 × 1
Pascal’s Triangle is a triangular array of numbers. Each next row has one more number than the last row.  Each row has 1's on both sides and every inner number is the sum of two numbers above it. It can span infinitely. The connection between mathematical combinations and Pascal’s Triangle is that the numbers in Pascal's Triangle represent mathematical combinations. If you sum each row, you can see that each row has powers of base 2, beginning with 2⁰=1 and so on.

N.L., (2018) Pascal's Triangle with Powers with the base of 2 

Triangular Numbers

If you start with row 3 and start at 1, the diagonal contains the triangular numbers.

N.L. (2018) Pascal's Triangle with Triangular Numbers

In conclusion to this action project, I learned many things from this project. I learned more about how I can relate reflections to Pascal's triangle and I found different patterns in Pascal's triangle. Some challenges for me during this process were coming up with a question and answer about mathematical combinations and creating my own Pascal’s Triangle. I found completing this action project enlightening because I know I a lot of effort into this action project. Overall I enjoyed doing this action project and I also enjoyed this unit.

Geometry Reflections In Buildings and Structures

This action project is for my course "Prove it or Lose It" and unit on finding reflections in the real world.  We went over what reflections were, how to find reflections, talked about how we can show them in the real world. In this action project, I found reflections and I also found two different ways to measure or quantify the reflection using mathematical terminology. I looked for my geometric reflections in buildings and structures.

The first way to measure the reflections in buildings and structures is using angles.  In the screenshot below, you can see how this building has different angles on its structure. You can also see how I found the angles that the buildings made. To understand this reflection, we can look at the angle of where they are equal. That is an important property of geometric reflections.

O'Neill, McKenna. (2012) Two Buildings. Web: Blogspot 

Another way that you can measure reflections is to measure reflections in buildings and structures is using slope.  In the picture below, you can see window bars. Instead of measuring the angles of the lines, you can find the coordinates of the points on the ends of the triangles on the window bars and calculate the slope of each line.

NL. (2018) Window Bars.

Using the slope formula, we find:
SLOPE of AD: (-6.5+-1.6) / (8 - 2.7) = (4.9) / (5.3) = 0.92
SLOPE of ED:  (-6.5+-1.6) / (13.5-8) = (4.9) / (5.5) = 0.89

My slope is off by a decimal of 0.3 but we can see that the slopes of the two lines are about the same.

This action project was challenging for me because I had a difficult time figuring out how I was going to include some mathematical terminology to show and measure reflections. Overall I enjoyed this action project and I hope that my skills with measuring reflections have increased because of this project.