BLOG #4 1/19/2016

Hello, welcome to my next blog post! So this week in geometry class, we are learning about lesson 3-3 Proving Lines Parallel,  3-4 Perpendicular Lines, and 3-5 Slopes of Lines. These lessons can help us in our architect skills so that is why we are learning them. 

Part of the problems given to us was to determine whether this diagram provide enough information to conclude to a||b. The vertical to 125 degrees is also 125 degrees. But since it is the supplementary of 55 degrees, then they same-side interior angles. Based on the converse of same-side interior angles, then a||b.

We also had to draw a figure in which the line segment AB is a perpendicular bisector of XY. However, we had to make XY NOT a perpendicular bisector to AB. This is what I drew:

Based on what I drew, the line segment AB is a perpendicular bisector of XY. Meaning, AB intersects XY in 90 degrees while one of AB's point is also XY's midpoint. However, XY is not a perpendicular bisector because it does not intersect line segment AB nor divides AB in half.

For the next topic, we had to think about a ladder. Have you ever noticed that all the rungs of a ladder or parallel? Maybe you did, maybe you did not. But the question is, "Why?"

After drawing all the ladder, I came into a conclusion as to why the rungs of a ladder are parallel. According to Theorem 3-4-3, if two lines are perpendicular to the same line, the the two lines are parallel to each other. In the ladder, all the rungs intersects both lines in 90 degrees. Because they are perpendicular to the same lines, then the rungs in the ladder are parallel.

Next, I've been challenged with a problem. The slope of line AB is greater than 0 and less than 1. I had to write an inequality for the slope of a line perpendicular to line AB. After solving the problem, I got the answer m < -1.

After that, I had to determine whether two cards are parallel if they were to drive the same speed and traveled the same distance. This is what I drew:

I concluded that since they traveled the same speed and distance, then they must be parallel because they both make a line. 

This week, we've also learned about slope. In fact, there are two ways to find the slop of a line.

Based on this image, these are two ways to find the slope: the rise and run method, or the slope intercept formula.

After this, I had to draw two parallel lines and discuss how their slopes are similar. 

The two red lines are in different points, however, they have the same slope. 

Finally, I had to do something similar to the last photo, but this time with perpendicular lines.

In this photo, each perpendicular lines are in different points in each graph. Although, each line have the same slope.

Blog Entry #3

Welcome back to another blog entry! The lesson that I am learning about this week involves using properties of equality and congruence. I am learning to write justifications to prove that every step in an algebraic equation is valid. If you guessed that our lesson is about Algebraic Proof, then you were right! 

So the main purpose of this lesson is help us develop the skill of determining whether every step we make is valid using justification. Developing this skill not only will apply to geometry, but it's possible to use this skill in our everyday lives. 

What I am doing is using the definition of segment congruence and the properties of equality to show that all three properties of congruence are true for segments.

Segment congruence is when two segments have equal measures and length

Now based on the definition of segment congruence and the properties of equality, I have proved that all three properties are true for segments because it shows similar examples. Reflexive, symmetric, and transitive properties of equality and congruence have the similar examples. Therefore, segments that are congruent are equal to each other.

So now that I finished explaining the first question, I will now proceed to the second question. In this next one, I need to compare the conclusion of a deductive proof and a conjecture based on inductive reasoning.

First of all, the conclusion of a deductive proof is found through deductive reasoning, which uses actual facts to draw a conclusion. A conjecture is a statement believed to be true. Inductive reasoning is using patterns that you observe to prove that a statement is true. Based on what I observe from both the conclusion of a deductive reasoning and a conjecture based on inductive reasoning, I can conclude that both are different from each other. The conclusion of a deductive reasoning can either be true or false. If one of the statements of deductive reasoning is false or can be contradicted, then the conclusion of a deductive reasoning is wrong. The conclusion of deductive reasoning is also based on actual facts. The conjecture of inductive reasoning is based on fact pattern, not by actual facts through research. 

In conclusion, I have learned a lot from this lesson. Algebraic proof helped me learn more about deductive and inductive reasoning. And because of this, I have developed to use reasoning skills that I can use in my everyday life. 

 

Blog Entry #2

Hello, I am back. This is my 2nd blog entry. So our geometry teacher gave us an assignment involving biconditional statements and its conditional and converse on lesson 2-4 on our geometry textbooks. We were given three questions. In our first question, we had to write the definition of a biconditional statement as a biconditional statement. Next, we had to use the conditional and converse within the statement to explain why our biconditional is true. 

Definition of biconditional: A statement that can be written in the form "p if and only if q." 

Biconditional: A statement is a biconditional if and only if it is written in the form "p if and only if q." 

Conditional: If a statement is written in the form "p if and only if q," then it is a biconditional. (Truth Value: True)

Converse: If a statement is written in the form "p if and only if q," then it is a biconditional. (Truth Value: True)

In order for a biconditional to be true, both its conditional and converse must be true. Both the conditional and converse of the biconditional are true; therefore, the biconditional is true. 



In our second question, we had to use the definition of an angle bisector to explain what is meant by the statement "A good definition is reversible."

Definition of Angle Bisector: A ray that divides an angle into two congruent angles.

(The red line is the angle bisector)

The statement "a good definition is reversible" applies to the definition of "angle bisector." The definition of "angle bisector" can be written as a biconditional, conditional, and converse.

Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles. (Truth Value: True)

Conditional: If a ray is an angle bisector, then it divides an angle into two congruent angles. (Truth Value: True)

Converse: If a ray divides an angle into two congruent angles, then it is an angle bisector. (Truth Value: True)

All statements are true; In each statement, the form changes but still keeps its truth value. The definition of "angle bisector" is a "good definition" because it is "reversible."

And finally, our third question asks us to write the two conditional statements that make up the biconditional "You will get a traffic ticket if and only if you are speeding." Then we have to determine whether the biconditional true or false and explain our answer. 

Conditional: If you get a traffic ticket, then you are speeding. (Truth Value: False, because speeding isn't the only traffic violation you can do to get a traffic ticket.)

Converse: If you are speeding, then you get a traffic ticket, (Truth Value: False, because drivers only get traffic tickets when they are caught speeding. Not all drivers who speed get traffic tickets.)

The biconditional is false because both conditional and converse were false. In order for the biconditional to be true, the conditional and converse must both be true. But since both are false, then the biconditional is false. 

Cereal Box Project

Hello! My name is Jeff, and I am a sophomore at Mount Carmel School in Saipan. Here at Mount Carmel School, I currently have seven class, one elective class, and a few extra-curricular activities. So a few weeks ago, my geometry instructor, Mrs. Buenaflor, assigned us a cereal box project. Now you may be wondering, "What is a cereal box project?" In this project, we redesign a cereal box and use our handicraft skills to make it smaller, but can still hold the same amount of cereal as the original size.

The cereal box I've chosen was Cheerios. Looking inside the Cheerios box, I've noticed that the cereal did not fit well inside. It's almost as if there is an equal amount of cereal and air. Half of the box is basically empty. The cereal only fills in the bottom half of the box, while air fills in the top half.

I started thinking how trees were cut down to make these boxes. If half of the cereal box won't even be used, then half of the trees that factories cut down are goes to waste. This is not just! Factories harm the environment in many ways, so why would factories continue to make these boxes in the same design when half of it isn't even being used?

So, in order to solve this problem, I made a box that can be a lot more environmental friendly. The Cheerios box I redesigned is smaller than the original size, but can hold the same amount of cereal. To help the environment, factories should be doing the same and make smaller cereal boxes. Changing the size and shape of the Cheerios box can reduce excess packaging.

This new cereal box design can hold the same amount of cereal, while having a smaller size. Because of its new size and shape, it can occupy less space in shelves. Store counters can hold more cereal boxes if all the boxes were to be in this design. Also, the look of this cereal isn't the usual shape. We all know how cereal boxes look like, but this cereal box can catch the attention of shoppers because of it's new design and unique shape!

Making this new design required a ruler, Cheerios cereal box, and a calculator. To find the measurements of the new design, it took many attempts in the calculator. I took the measurements of the original box, then I calculated it's surface area and and volume. To find the measurements of the new design, I used the same volume, but a smaller surface area. The relationship between the surface area and the volume is: without the surface area, you can't determine if the new design occupies less space than the original. Without the volume, you can't determine whether the box can hold the same amount of cereal or not. Making the new design required the calculations for both surface area and volume.

In conclusion, this new cereal box is better for the environment. This project assigned by my geometry teacher help me learn not just about surface area and volume, but also how we can make this factory-made product a lot more environmentally friendly. So to help the environment, we should make all Cheerios cereal boxes in this new design. It's new design is both appealing to an shopper's eye and friendly to the environment.