Algebraic Proof: Blog Entry #3

Throughout this blog entry, I will be identifying some of the properties of equality and the properties of congruence and reviewing them to use and write them as algebraic proofs.

Proof uses logic, definitions, properties, and proven statements to show that a conclusion is true. When writing proofs, it’s important to give justifications to show that each step is valid. These justifications can be made by using any piece of information that can be obtain from a problem.

Solving an equation uses proof, because several properties are used to solve the equation. Using  the Properties of Equality can help prove a statement to be true.

Example:

 Segment congruence means two line segments are congruent if they have the same length.

By knowing this, we could come to the conclusion that each property of equality have their own corresponding property of congruence. Properties of equality do apply to segments as well, and also three properties of congruence does too.

Reflexive Property of Equality: Segment AB = Segment AB

Reflexive Property of Congruence: Segment AB ≅ Segment AB

Symmetric Property of Equality: If Segment AB = Segment CD, then Segment CD = Segment AB

Symmetric Property of Congruence: If Segment AB ≅ Segment CD, then Segment CD ≅ Segment AB

Transitive Property of Equality: If Segment AB = Segment CD and Segment CD = Segment EF, then Segment AB = Segment EF

Transitive Property of Congruence: If Segment AB ≅ Segment CD and Segment CD ≅ Segment EF, then Segment AB ≅ Segment EF

  From the definition of segment congruence, the properties of equality and the properties of congruence are the same. It is true since they prove to be the same to each other, and the definitions, statements, and examples of each property are the same. We were able to  prove that these properties are true for segments .

 A deductive proof uses logic and reasoning to come to a valid conclusion. A conjecture based on inductive reasoning, is a statement believed to be true based on patterns formed from multiple observations. Using deductive reasoning and inductive reasoning is different, the processes use different methods to achieve solutions.

Blog Entry #2

35.) Write the definition of a biconditional statement as a biconditional statement. Use the conditional and converse within the statement to explain why your biconditional is true. 

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 statement. (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

The biconditional is true since both the conditional and the converse are true.

36.) Use the definition of an angle bisector to explain what is meant by the statement "A good definition is reversible."

Definition: An angle bisector is a ray that divides an angle into two congruent angles.
A good definition is reversible since its statement has true value as a conditional statement and as a converse.
Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles.

Conditional: If a ray is an angle bisector, then it divides an angle into two congruent angles; True

Converse: If a ray divides an angle into two congruent angles, then it is an angle bisector; True

The definition of an angle is a good definition because it is reversible. Both conditional and converse are true, so that means the whole definition is true. 

41.) Write the two conditional statements that make up the biconditional "You will get a traffic ticket if and only if you are speeding." Is the biconditional true or false? Explain your answer.

Conditional: If you get a traffic ticket, then you are speeding.
Truth Value: False; you can get a ticket for other issues, like passing a red light.

Converse: If you are speeding, then you get a traffic ticket. 
Truth Value: False; You only get a traffic ticket when caught speeding. Those who don't get caught speeding does not get a traffic ticket.

The biconditional is false because both conditional and converse were false.
For the biconditional to be true, the conditional and converse have to both be true. But since they are both not true, the biconditional is false.

Redesigning Cereal Box

Hi! My name is Erika Mendiola but you can call me Erk for short. I am currently 14 years old, turning 15 this November 27th :) I love cooking, baking, & reading books. I am a Sophomore at Mount Carmel School, Saipan. This is my blog for Mrs.Buenaflor's 4th period Geometry class.

A few weeks ago, my class was given a project called, Cereal Investigation: Maximizing Volume, Minimizing surface area. We had to calculate all the dimensions of the cereal box and create a smaller box that could still fit in the cereal.

The cereal box that I used was Cheerios. What I noticed about the package was that half of it was unnecessary. The cereal box had half of Cheerios (which was on the bottom of the box of course,) and half air which filled the top.

If you think about the cereal box and how it was made, you'll know that the box was made from trees. If all Cheerios cereal, and other cereals, wasn't filled with half of air, think about how many trees that would've been saved from being cut down just to make the cereal box when half of it was really unnecessary. Making the cereal box smaller would save money, trees, and it would be very economically friendly.

What I, and everyone, could do to change the packages to reduce the excess packaging is to find a way to redesign a smaller cereal box, that could fit the same amount of cereal into it. I believe that the factories that make the cereal boxes should do this too.

The new cereal box I made could hold the same amount of Cheerios. It saves up space since it is smaller, and if cereal boxes were smaller, people would be amazed at how small the box is and how much cereal it holds!

Making the new cereal box took me many tries till I finally got it right. Without the surface area, you would not be able to find out if the new cereal box design takes up less space than the original cereal box design. And without the volume, you can't tell if the new box could hold the same amount of cereal or not.

Redesigning a new cereal box was very frustrating for me. But it did make me realize that the cereal boxes that factories made today are not economically friendly. If factories redesigned cereal boxes, it would save money, time, and trees. It would also be much better for the environment.



This is my redesigned cereal box: