Algebraic Proof

     Hey there! This week's topic is about algebraic proof. I am reviewing and identifying properties of equality and then using them to write algebraic proofs. A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. 

     The definition of a segment congruence: two segments that have the same length. The reflexive property of equality corresponds to the reflexive property of congruence. It can be written as A equals A or A is congruent to A. They are the same thing.  Moving on to symmetry, the symmetric property of equality and the symmetric property of congruence are similar as well. For equality it can be written as "a=b, b=a", for property of congruence it can be written as "if a is congruent to b, then b is congruent to a". Last but not least, transitive property of equality and congruence. In transitive property of equality, you can write "1=2, 2=3, 1=3". It would mean the same thing as saying, " 1 is congruent to 2, 2 is congruent to 3, 1 is congruent to 3". They both mean the same. 

     I'm going to use the conclusion "If the first letter of the alphabet is A, then the second letter of the alphabet is B." This conclusion uses facts and patterns. Also, if this were a conjecture, you would know that the third letter of the alphabet is C. You know it's C because of the pattern and direction the blogger is going in. 

     We've actually already done proof! Can you believe it? The equations we learned to solve in 8th grade is an example of when we learned to apply proof. For example, in different equations we learn how to use distributive property and substitution property. When you justify your steps of an equation it is a way of validating and authenticating your steps. 

The pictures were taken out of my textbook. They are not mine. No copyright intended.

Biconditional Statements

     Hello! For this week I will be talking about biconditional statements. You might be asking yourself, "What exactly is a biconditional statement?" To answer your question, a biconditional statement is statement that can be written in the form "p if and only if q." A biconditional statement is when you combine a conditional statement and it's converse. 

     I answered pages 100-101 and did numbers 35,36, and 41. For #35, the biconditional statement is: A statement is a biconditional statement if and only if it can be written in the form "p if and only if q." I took the definition of a biconditional statement and then wrote it in the form of a biconditional statement. The conditional statement is: If a statement is a biconditional statement, then it can be written in the form "p if and only q." A conditional statement is written in the form "if p, then q".  For the converse: If a statement can be written in the form "p if and only if q", then it is a biconditional statement. The converse is when you reverse the conditional statement, it is written in the form "if q, then p". Since both the converse and conditional statements are true, the biconditional is true. 

#36 .) Definition: a ray that divides an angle into two congruent angles. The conditional statement is: if a ray is an angle bisector, then it divides an angle into two congruent angles. This conditional statement is true. Now the reversed version or converse is: If a ray divides an angle into two congruent angles, then it is an angle bisector. The converse is also true. A good definition is one where it is either reversed and forward and you still have a true value. 

#41.). The 1st conditional statement: if you get a traffic ticket, then you are speeding. This conditional statement is false. The 2nd conditional statement: if you are speeding, then you get a traffic ticket. This 2nd conditional statement is true. If you want your biconditional to be true, your conditional and converse both have to be true as well. But here, since the conditional statement is false, the biconditional is also false. The 1st conditional statement is false because you can get a traffic ticket for going to slow or violating other laws, not just speeding. 

     A good definition is able to be used either forward or backwards. A biconditional statement requires your ability to write a conditional statement and a converse. If you are able to properly write a conditional and converse,then you are a step closer to writing a biconditional statement. The biconditional statement also test your ability to think and look at different perspectives.

 

Pictures of biconditional statements:

 

1.) How to write biconditional statements symbolically and using words:

2.) Examples of biconditional statements:

Redesigning Cereal Box

Hello! My name is Theresa Zheng, and I am 15 year old. I am currently enrolled at Mount Carmel School (best decision I've ever made). I am a cheese and pug enthusiast. 

The students were given a project called redesigning a cereal box. The main objective of this project was to minimize surface area while maximizing the volume. I had to create a new cereal box where the original amount of cereal would be able to fit in.

     The package was very sturdy because of the cardboard packaging. Pulling the cereal out of the box, I notice that there was minimal space, therefore, the cereal had to come out from a tight space. The products inside were only filled halfway in their plastic bags. Producers should really fill their cereals all the way. The cereal was just ordinary flakes with dried strawberries added to it. 

     Yes! The cereal box is made out of recycled cardboard. Companies, like Kellogg's, have actually been making their packaging out of recycled cardboard boxes. If you look under the box it even promotes recycling and being environmentally friendly. Being a environemntally friendly box attracts new customers. 

     I would make it thinner. The cardboard box doesn't have to be too think. Also, since the cereal inside the plastic packaging isn't all the way to the top, I would reduce the height of the box. This way everything is smaller and more environemntally friendly.

     The box design affects the amount of space it would take on a shelf. When people see the cereal box in stores, they decide if they should get it not only by the flavor, but by the size of the box. I mean, what if they get home and it can't fit in their pantry? No one wants a big and bulky box. 

     The amount (volume) of cereal you have must be able to fit in the cereal box (surface area). Whether it is the size or height, it must fit. When solving, the new and old surface area will turn out differently from each other, but the volume stays the same. The amount of cereal can fit in either the original or new box. 

     I was befuddled at first. I had to sit back and really try and play around with the numbers (length, width, and height). I tried many times and sooner or later I found the solution! The original and new surface area were different, but the volume for both was the same. The new surface area was a smaller amount. It was challenging but not impossible!