The Proof

For this blog, I am going to complete the ‘doing math’ requirement.

Euclid didn’t just develop the concepts of proofs, he also developed a set of conjectures and propositions that connected all proofs together and allowed them to make sense while looking at them piece by piece. Euclid used the most general concepts to create these proofs that explain larger theories, in an attempt to prove the proposition or conjecture correct. In the process of explaining these theories, we started with proving theorems. Theories consist of a set of ideas that are used in order to explain why something is true, while a theorem is a result that can be proven to be true using axioms or postulates. We have proven our theories and shown that they are true through proving theorems and conjectures, showing that the invention of the proof is more important than we think. In the process, he uses complicated wording and bases most of his geometric proofs off of his five postulates.euclid_postulates

Euclid’s five postulate, representations shown above, consist of: “1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center. 4. All right angles are congruent. 5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. The fifth postulate is known as the parallel postulate ” (Wolfram Alpha). Euclid did not just use these five postulates while creating the idea of the proof and proving different theories, he also had a set of common notions.
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These common notions were used to guide the rules of completing proofs and what should commonly be followed by others when attempting to do so. Since Euclid has created the concept of a proof, things have changed. Euclid brought the idea of proofs into the world of mathematics, showing that there is a strategic and more sophisticated way of proving an idea. Euclid set the groundwork for bigger and better things that occurred later on in the mathematical world.

Since Euclid has created proofs using conjectures, postulates, and theorems, proofs and mathematics have come a long way. Since math has continued to develop, the list of ways to prove things has advanced. We have gone beyond the idea that geometry is the only thing that can be proven using axioms and postulates. We have advanced into more complicated areas of mathematics, like proving that an equation can be solved or proving that something equals something else. Euclid primarily focused on geometry, leaving so much more room for improvement. For example, the creation of the proof and Euclidean geometry lead to the creation of Hyperbolic and Elliptic geometry.
550px-TriangleGeometry
Although he only focused on geometry, he did get a lot of things right. The idea that there needed to be a basic list of axioms or postulates in order to start, set the framework for the concept of writing proofs and was the biggest improvement in the world of mathematics. Euclid even set up the perfect outline for writing proofs, starting with what you know and ending with what you wanted to prove. To this day, this is still the general outline for how proofs should be written. The idea of combining multiple conjectures in order to create one proof through valid reasoning is another thing that Euclid did right, the need for explanations is essential while writing a proof. If the reasoning is correct, there is no room for argument, therefore showing that the reasoning about why the conjecture is correct serves as a valid proof.

Now that all I’ve done is talk about Euclid and his works, now we should check out just how some of it works. Many of the conjectures and theorems could be proved just using the five postulates or the axioms, which was very interesting! I am going to prove a theorem doing exactly that:

Consider the following incidence geometry defined by these three axioms:

A1. There are exactly four lines in the geometry.
A2. Given any two distinct lines, there exists exactly one point that lies on both of them.
A3. Every point is on exactly two lines.

Theorem: Each line in this geometry contains exactly three points.

Proof. 

According to the first axiom there are exactly for lines in this geometry. Call these four lines l, m, n, and o. Take line l. By the second axiom, lines m, n, and o must each have a point in common with line l. 

The intersection of these three lines (m, n, and o) with l, create three distinct points at their intersections. The reason that these points are distinct is because if they were not distinct they would violate the third axiom, creating a point on three lines.

Line therefore contains three distinct points at the intersection with line m, n, and o. 

Suppose we have a fourth point P on line l. P cannot be on line m, n, or o because doing so would violate the second axiom. But every point is on exactly two lines, thus we would need a fifth line in order for point P to exist. But this violates the first axiom that only allows for four lines in the geometry. Thus we cannot add a fourth point to line l. 

Since we cannot add another point to line  and it must have at least three points at the intersections with m, n, and o, l must have exactly three points on it.

Similar arguments can be made for lines m, n, and o. Therefore, each line in this geometry contains exactly three points.                                           QED.

Euclid also was able to prove the pythagorean theorem in many different ways using the proof! I am going to take the pythagorean theorem and prove it only one way but there are many!

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Theorem: If triangle ABC is a right triangle, then the measure of angle c will equal 90º, b = |AC|, c = |AB|, and a = |CB|, then a² + b² = c².

One of the many ways of proving the pythagorean theorem includes a square inscribed within a square. Screen Shot 2015-11-28 at 12.57.12 PMWe can prove this algebraically, which makes it much easier to see. First, we can separate this into the square contained on the inside, whose area is c². Next, we have 4 triangles located on the outside of the square, all with area ¹⁄2ab. Lastly, we have the area of the big square, which is (a+b)². Now, the are of the large square must equal the area of the small triangles and the smaller square contained within so we can set these numbers equal to each other…

c² + 4(1/2 ab) = (a+b)²
c² = (a+b)² – 4(1/2 ab)
c² = a² + 2ab + b² – 2ab
c² = a² + b²

This allows us to see why the pythagorean theorem works and it proves the pythagorean theorem at the same time!

I believed that proof mattered to Euclid because it was a solid base for the studies and work that he created for Euclidean geometry. He was working with geometry, attempting to expand on what he was finding. In order to expand, there needed to be a base, something to work up from. Proofs were important to Euclid because it shows that his work was expanding, and it shows that his work is valid. He created the framework for proofs, and this became more and more important to him as he began to make more and more discoveries about how complicated geometry is. But when you think about the history of math and the advancements that were made, you can ask yourself, why did Euclid create the proof? In my opinion, I think that he created the proof because there was no other formal way of completing mathematics. I think that Euclid wanted a concrete format for proving theorems. If these theorems were proved, they would be able to log them and build the world of mathematics on top of them instead of continuing to work with ideas that may or may not be true. To this day, proofs are the most important framework for figuring out whether ideas are true or not. Mathematicians and even scientists work to prove theories and new ideas through already proven ideas, if the framework was not set through these proofs then there would not be the advancements that we have today.

 

What is Math?

When thinking about math and how all the equations and concepts were discovered, it makes one wonder how all of it came together to form the subject of math. When I think of math, I think of numbers. This is one of the greatest discoveries in the history of math, the creation of numbers and the application of these numbers into other things. For example, using these numbers to add or subtract. Using these numbers for counting, measurements, anything that was possible to use numbers for lead to the biggest discovery in the history of mathematics. Numbers are the foundation of math and the ability to extend math into other areas. Next, I would think that the biggest discovery is measurement. The ability to take these numbers and apply them to measuring certain objects, angles, lengths, what have you, would be another huge discovery in the area of mathematics. This is huge because it was the foundation for geometry, the equations involved in geometry, measuring simple lengths, constructing buildings, cooking, chemistry, measurement and numbers take the cake for the greatest top two discoveries in the history of mathematics.

Other than numbers and measurement, the creation of geometry was a milestone for the history of mathematics. They had all the components of geometry but putting it all together was the biggest part of it all. After the creation of geometry and fully understanding how geometry worked, it lead into other aspects of geometry other than just Euclidean. For example, it lead into Hyperbolic geometry and Elliptic. Both being very abstract forms of geometry that couldn’t have been discovered without the creation of Euclidean geometry.

From these discoveries, other types of abstract math started branching off of each other. Math started introducing topics like statistics, modern algebra, discrete mathematics, but they were all introduced based on the common idea of numbers and measurements. How the different types of numbers like, rational or irrational, produce different answers and how they work in different planes or environments. The idea that math can go into different dimensions changed the world of mathematics because now mathematicians don’t just work in 2-D, but 3-D or 4-D. Without the simple discovery of numbers or measurement, we might not be as advanced as we are now in the world of mathematics or technology.

Lesson on Chance

For this lesson, students will be introduced to the idea of chance. The teacher will set up five stations of chance, each with a game to play for students to interact with. First, the teacher will talk about probability and explain the formula for finding probability. The teacher will go through examples of probability and explain what different fractions would mean in terms of the chance situation. Students will go through experiments in order to gain a conceptual understanding of chance and probability. For example, they will find the chances of flipping heads, getting a 4 on the die, picking a red card, picking a diamond, and picking the 5 of diamonds. Through experimentation, students will understand what probability is and how to compute the probable outcome. In addition to understanding probability, the students will understand the difference between unlikely, most likely, and very likely when determining their probabilities. A link to this lesson is posted below:

http://illuminations.nctm.org/Lesson.aspx?id=2895

This lesson aligns with the CCSSM standards for grades 6-8 because it is an introduction into probability and what it means. It allows the students to gain a greater understanding of how probable an outcome is and what the chances of it actually happening are based on experimentation. They can draw conclusions about the probability of events, overall I would give the lesson a rating of 5.

Teaching Portfolio – Lesson Two

Opening this lesson with a review, we asked the students what they remember about mean as a fair-share and the mean as a balance point. Through discussion, we had the students talk about what they remember and how they would express both of the representations. We then gave the students 8 different distributions of a data set, all with the same mean. For convenience, these distributions were all cut out prior to the lesson in order for the students to be interactive with the lesson. After the students were given these 8 distributions, we asked them to put them in order from least variable to most variable by completing a visual observation. Once the students realized that this was difficult to complete, we introduced the total absolute deviation, also known as the TAD. At first, we showed an example of how to compute the TAD using one of the distributions and then instructed the students to complete these computations for each of the other 7 distributions. We explained that the numerical value represented the total distance each data point is away from the mean. We then asked the students to rank these distributions again based on variability. After they completed this, we then asked how the students would be able to rank these distributions if they had data sets with different means and a different number of data points. Through guided instruction, the students were able to determine that the MAD, or mean absolutely deviation, would provide better results. We computed the MAD for one of the distributions and then had the students compute the MAD for the other 7 distributions. Once again, they were asked to rank them in order of least variable to most variable, explaining that the MAD was the average distance each point was away from the mean. Wrapping up this lesson, we were able to have a discussion about how the mean is an important measure and without it, it would be difficult to compute how variable a data set is. For review, we went over the TAD and the MAD again, ensuring that the students had a conceptual understand of what these measures represented. Below, I have attached the distributions that we cut out and used in order to rank variability. I have also attached the lesson outline for the MAD, this was a lesson that was created by Jon Hasenbank and Tara Maynard, I used this lesson to teach the concepts with some modifications.

Redistributions

_Lesson Outline MAD

This lesson aligns with the CCSSM grades 6-8 standards due to it’s connection with variability. The focus of the lesson was for students to understand how the TAD and the MAD account for variability, and eventually can be used to make conclusions about data sets. Overall, this lesson would be rated as a 4. It allowed the students to be introduced to the TAD and the MAD, which they will use later on, but I feel as though it should be focused more on the MAD and how this measure is important in terms of variability. I think it would be beneficial to include data sets with different means and number of data points in order to show how the MAD is a more powerful measure and how it determines variability.

Teaching Portfolio – Lesson One

Mean as balance point and fair-share: The goal of this lesson was for students to understand the mean as a balance point and fair-share. To begin this lesson, we first asked each student what their interpretation of the mean is with the intention of creating a discussion. To see that students had a conceptual understanding of the mean, after they all explained their interpretation, they were then asked to compute the mean for a random set of data. Our main activity consisted of finding data as a group, we asked how may people they have in their immediate family. This data is collected through a quick survey of our group members, with a total of six pieces of data for our collection. You are able to adjust the number of people in certain families in order to get a “nice” mean if you wanted to. Through manipulatives, the students can explore the different ideas of balance point and fair-share. For example, we used Unifix cubes in order to demonstrate the different representations of the mean. After we showed examples of the fair share, we asked the students to do the same thing by exploring with the manipulatives. We asked them questions in order to get them thinking about how the fair share is represented. We did the same with the balance point as well, we showed examples and then let the students gain a greater understanding through their exploration with the cubes. The format of our lesson is included below:

Lesson Plan 1 draft 3-3

My rating of this activity would be a 5, where 1 is low and 5 is high. This activity was able to determine the difference between fair share and balance point. The students were able to see how the balance point is shown on a graph, but also how you can find the mean by the concept of a balance point on a graph. When introducing fair share, the students were also able to see how combining all of the data points with the cubes and then redistributing them into equal points gives a representation of the mean. Overall, this activity allows the students to gain a greater conceptual understanding through experimentation as well as observing the examples to illustrate the differences. Relating to the CCSSM Probably and Statistics Standards for grades 6-8, this lesson allows for a student to understand a measure of center and summarize a data set in relation to it’s context. When referring to the GAISE report, these students align with this report’s recommendations because truly understanding the data and the mean, with a deeper understanding of how the mean can represented, is the first step of moving a Level A student to a Level B student.

Reflection

From teaching at Zeeland, I learned that students could know more than you think they know and they need to be presented with the right tools in order to show it. On the assessment that we were given, most of the students answered some of the questions incorrectly. But while going through the lesson and giving students manipulatives to work with, they showed their true knowledge of the subject. The students were able to absorb the information they were presented with pertaining to fair share and balance point easily as well as show the knowledge they had been taught. I learned that all students need an opportunity to speak and share their knowledge or else they could potentially be left behind. These things will impact my practice because each students needs an opportunity to show their knowledge as well as expand on it. I will be sure to present students with opportunities to grow in each lesson using manipulatives or other materials to ensure proper learning. Next, I want to continue to learn about how to help students expand on their knowledge and become proficient on topics that I will be teaching them. I want to learn how to teach the mean and the MAD easily for students to have a greater understanding without confusion.

Learning Focus for Students

To teach students on Thursday, my personal focus was to really focus on fair share and balance point when talking about the mean. The students in the group I was assigned to seemed to be on the right track when analyzing the assessment questions they had answered the previous day. This tells me that they know how the mean works, or at least how to compute the mean. This leads me to believe that the goal I should keep in mind is moving forward past fair share and introducing the idea of balance point. The goal is for students to understand how balance point works, how they can maintain the same mean with moving points around on a number line. Also, how they can find the mean by using the number line as a balance point. If things are going well for the lesson, I would like to see students actively participating and answering questions that are asked. I would like to see students forming different data points on the number line in order to keep the same mean. The greatest thing would be to actually see students producing a real conceptual understanding where they can come up with their own data sets and compute the mean using a balance point as well as distribute data points easily to show fair-share. To prepare for this, I would like to have an idea of what the lesson should look like. I would like to have an idea of a general mean that all the data points should add up to in order to get an easier mean. It would help me to go over my lesson with someone who doesn’t understand the subject as well in order to prepare myself for questions that the students might ask.