## Unit One Portfolio:

## Cover Letter:

The Pythagorean Theorem & Coordinate Geometry:

During this unit I learned more about the actual relationship between coordinates and the reactions of different shapes, angles, lines, and forms of change. The first area of math that we looked at was right triangles and how pythagorean theorem was formed and can be applied in different ways. Another was the process of locating, observing and understanding different points on the coordinate plane. At the beginning of the unit, when we were just getting introduced to coming up with proofs and problem solving, we were able to solve where the distance and midpoint formula came from which are Distance= =√((x_2-x_1)²+(y_2-y_1)²) and for midpoint=Add both "x" coordinates, divide by 2. Add both "y" coordinates, divide by 2. After being given the problem with highways, we saw how angles changed based off of one-another, which led into our further investigation of the relationship between lines and lengths. In the second POW, I mentioned the idea of developing these formulas, but I was still not positive on how important they were. In “Proving With Distance Part II” you can see a more developed, proper use of these formulas and how they are crucial in finding out solutions in coordinate geometry (and also connecting that to circles too!)

Lines, Angles, and Other Points:

What I found to be extremely useful during this unit was learning about working with little to no given units. In geometry, there are many times where you have to work with little information. In order to be able to solve problems, you need to observe and understand concepts without measurements. Especially at the beginning of the unit where we were given the unit problem, we knew little information but were still asked to make many predictions. There are three very important areas of geometry that we looked at: Finding the distance from points to lines, line bisectors, and angle bisectors. A representation of each of these can be seen in the portfolio selection of work (“POW 1”, “Another Kind of Bisector”, “portfolio work 2”) each of these was crucial in finding the final answer to the unit problem. Also, the development of these skills helped us in understanding true proof.

Circles & the Square-Cube Law:

The square cube law explains the relationship between shapes and when they undergo change in size. More specifically, when a shape goes through an increase in size, the surface area will increase slower than the volume. This relationship is not linear,. During this unit, we also explored the difference between volume and area. My first development of these ideas came from two different activities. During in-person lessons, we had to explain the difference between the two. Area represents the more 2d planes of an object while volume shows how much space a 3d object will occupy. My understanding of the law itself came from the assignment, “The Square Cube Law” (labelled as “Portfolio Work) where we had to solve for different units of measure such as volume and surface area and explain how each object reacted. Basically, I saw where the law came from. For the unit problem, we had to understand the increase in area over time so we could understand the answer to the unit problem. We had to apply our knowledge of the square cube law and formulas of shapes to know that each trunk will grow a certain amount each year.

Proof:

One of the largest portions of what I learned in this unit is the approach of proof. When you usually think of proof, you think of pieces of evidence that can display the work of a problem and solution. Though, the difference with proof is that it shows the true process of arriving at conclusions. The reason it is so difficult is because you need to answer all the hows and whys in the process. During the unit we were taught how to properly show all our work and how important it is to know how each system and step of a solution works. In each problem, it is important to include: an analysis of the problem being asked, observing and taking in each part of the question, then answering each part and fully understanding and showing how I got there. Especially when it comes to geometry, there are a lot of times when you need to fill in the gaps or find different units, it is extremely important to know how to prove these units! Like the data-primer assignments (labelled as “portfolio work”) we had to observe given information and understand where they came from and how we could get there on our own. Being able to do this with others work as well as our own

During this unit I learned more about the actual relationship between coordinates and the reactions of different shapes, angles, lines, and forms of change. The first area of math that we looked at was right triangles and how pythagorean theorem was formed and can be applied in different ways. Another was the process of locating, observing and understanding different points on the coordinate plane. At the beginning of the unit, when we were just getting introduced to coming up with proofs and problem solving, we were able to solve where the distance and midpoint formula came from which are Distance= =√((x_2-x_1)²+(y_2-y_1)²) and for midpoint=Add both "x" coordinates, divide by 2. Add both "y" coordinates, divide by 2. After being given the problem with highways, we saw how angles changed based off of one-another, which led into our further investigation of the relationship between lines and lengths. In the second POW, I mentioned the idea of developing these formulas, but I was still not positive on how important they were. In “Proving With Distance Part II” you can see a more developed, proper use of these formulas and how they are crucial in finding out solutions in coordinate geometry (and also connecting that to circles too!)

Lines, Angles, and Other Points:

What I found to be extremely useful during this unit was learning about working with little to no given units. In geometry, there are many times where you have to work with little information. In order to be able to solve problems, you need to observe and understand concepts without measurements. Especially at the beginning of the unit where we were given the unit problem, we knew little information but were still asked to make many predictions. There are three very important areas of geometry that we looked at: Finding the distance from points to lines, line bisectors, and angle bisectors. A representation of each of these can be seen in the portfolio selection of work (“POW 1”, “Another Kind of Bisector”, “portfolio work 2”) each of these was crucial in finding the final answer to the unit problem. Also, the development of these skills helped us in understanding true proof.

Circles & the Square-Cube Law:

The square cube law explains the relationship between shapes and when they undergo change in size. More specifically, when a shape goes through an increase in size, the surface area will increase slower than the volume. This relationship is not linear,. During this unit, we also explored the difference between volume and area. My first development of these ideas came from two different activities. During in-person lessons, we had to explain the difference between the two. Area represents the more 2d planes of an object while volume shows how much space a 3d object will occupy. My understanding of the law itself came from the assignment, “The Square Cube Law” (labelled as “Portfolio Work) where we had to solve for different units of measure such as volume and surface area and explain how each object reacted. Basically, I saw where the law came from. For the unit problem, we had to understand the increase in area over time so we could understand the answer to the unit problem. We had to apply our knowledge of the square cube law and formulas of shapes to know that each trunk will grow a certain amount each year.

Proof:

One of the largest portions of what I learned in this unit is the approach of proof. When you usually think of proof, you think of pieces of evidence that can display the work of a problem and solution. Though, the difference with proof is that it shows the true process of arriving at conclusions. The reason it is so difficult is because you need to answer all the hows and whys in the process. During the unit we were taught how to properly show all our work and how important it is to know how each system and step of a solution works. In each problem, it is important to include: an analysis of the problem being asked, observing and taking in each part of the question, then answering each part and fully understanding and showing how I got there. Especially when it comes to geometry, there are a lot of times when you need to fill in the gaps or find different units, it is extremely important to know how to prove these units! Like the data-primer assignments (labelled as “portfolio work”) we had to observe given information and understand where they came from and how we could get there on our own. Being able to do this with others work as well as our own

## Solution:

Introduction:

For this unit problem, we were given a story about two individuals (Maddie and Clyde) who plan on planting an orchard of trees on their new property. Their main goal is to be able to have a place to “hideout” in the very center of their orchard where they plan on not planting a tree-hence the name of the unit. For this hideout, they want it so that they cannot see in or out from the center point. In order to solve this problem, we were given the information that the orchard is 50 units in radius (it's a circular orchard.) With that, we needed to understand what a unit represented, the area of the trees, their growth rate, and where our last line of sight would be.

Process & Justification:

First, to solve the problem, I gathered all the necessary information. Again, through the unit, we learned: The orchard has a radius 50 units, one unit is equal to 10 feet, the last line of sight will pass through coordinates (25,½) and pass through (50,1),the starting circumference of each tree will begin at 2.5 inches and will increase by 1.5 inches in area each year. The next step can be easily understood in the following diagram....

We want to know where and when the trees will grow to the point of the line of sight, because when the two trees reach it, it will become a true hideout; the answer to the unit problem. In the diagram, you can see two trees on either side of the line of sight labelled c and d and two triangles labeled a and b. We know that both triangles are similar because they both share an angle and are right (meaning they have a 90 degree angle) This means that because we know the measurements of one triangle, we can find the units of the other. Once we find the short leg of triangle b, we will have the radius of circle 6. “How do we know that this is the radius of the circle?”,”it’s because the hypotenuse of triangle A is tangential to the circle! Plus, the center point of the circle falls on the long leg of this triangle, which is similar to b!” So, let's find the hypotenuse of A so that we can find the measures of B. Using pythagorean theorem, the hypotenuse is the square root of 2501 or approximately 50.0099990002. Now that we have this, we can solve for the radius or short leg. (seen below) ...

Now we have discovered that the radius is 0.019996 units. Now, one unit equals 10 feet so therefore the radius is 2.4 inches. Because we know the final radius, we can find the final area of the tree. Using the equation for area we get A= 18.0883 in^2. The next important step is to apply the starting area of the trees. If we begin at a circumference of 2.5 inches, we can extract the radius by reversing the formula for circumference;which is 2 x pi x r. So, 2.5 = 2 x pi x r. Whis gives us a radius of 0.398 inches. Apply that to the formula for area and you get A=0.497 in^2. Finally, we can subtract the starting area from the final area to see how much the trees need to grow (which is 17.591 in^2) and divide that by the amount of time it will take (17.591 in^2 divided by 1.5 in^2) Through which, you will get the final solution, 11.73 years.

Solution:

After planting all their trees in their orchard that is 500 feet in radius, Maddie and Clyde would get a true hideout after 11.73 years or 11 years, 8 months, 22 days, 4 hours, 46 minutes, and approx 36 seconds!

For this unit problem, we were given a story about two individuals (Maddie and Clyde) who plan on planting an orchard of trees on their new property. Their main goal is to be able to have a place to “hideout” in the very center of their orchard where they plan on not planting a tree-hence the name of the unit. For this hideout, they want it so that they cannot see in or out from the center point. In order to solve this problem, we were given the information that the orchard is 50 units in radius (it's a circular orchard.) With that, we needed to understand what a unit represented, the area of the trees, their growth rate, and where our last line of sight would be.

Process & Justification:

First, to solve the problem, I gathered all the necessary information. Again, through the unit, we learned: The orchard has a radius 50 units, one unit is equal to 10 feet, the last line of sight will pass through coordinates (25,½) and pass through (50,1),the starting circumference of each tree will begin at 2.5 inches and will increase by 1.5 inches in area each year. The next step can be easily understood in the following diagram....

We want to know where and when the trees will grow to the point of the line of sight, because when the two trees reach it, it will become a true hideout; the answer to the unit problem. In the diagram, you can see two trees on either side of the line of sight labelled c and d and two triangles labeled a and b. We know that both triangles are similar because they both share an angle and are right (meaning they have a 90 degree angle) This means that because we know the measurements of one triangle, we can find the units of the other. Once we find the short leg of triangle b, we will have the radius of circle 6. “How do we know that this is the radius of the circle?”,”it’s because the hypotenuse of triangle A is tangential to the circle! Plus, the center point of the circle falls on the long leg of this triangle, which is similar to b!” So, let's find the hypotenuse of A so that we can find the measures of B. Using pythagorean theorem, the hypotenuse is the square root of 2501 or approximately 50.0099990002. Now that we have this, we can solve for the radius or short leg. (seen below) ...

Now we have discovered that the radius is 0.019996 units. Now, one unit equals 10 feet so therefore the radius is 2.4 inches. Because we know the final radius, we can find the final area of the tree. Using the equation for area we get A= 18.0883 in^2. The next important step is to apply the starting area of the trees. If we begin at a circumference of 2.5 inches, we can extract the radius by reversing the formula for circumference;which is 2 x pi x r. So, 2.5 = 2 x pi x r. Whis gives us a radius of 0.398 inches. Apply that to the formula for area and you get A=0.497 in^2. Finally, we can subtract the starting area from the final area to see how much the trees need to grow (which is 17.591 in^2) and divide that by the amount of time it will take (17.591 in^2 divided by 1.5 in^2) Through which, you will get the final solution, 11.73 years.

Solution:

After planting all their trees in their orchard that is 500 feet in radius, Maddie and Clyde would get a true hideout after 11.73 years or 11 years, 8 months, 22 days, 4 hours, 46 minutes, and approx 36 seconds!

## Reflection:

Mathematical Reflection:

“How did the unit improve your understanding of the relationship between algebra and geometry?”

The transition from algebraic math to geometric can be difficult, especially when applying new and old methods. It's important to understand each of the subjects and to review them before understanding pre-calculus methods and beyond; they both create a base for learning.For me, personally, I see algebra as a way to dig deep into graphs, formulas; Geometry seems to be more about shapes and understanding space and angles within them. Even if that is not what they are truly about, to me I find that the best way to categorize them. During the unit, we were mainly focused on real proof, not just the problem and answer. Learning how to fully prove something is a good way to understand the relationship between these two concepts. Once I was able to fully understand and actually prove a concept during the unit, it made it more valuable and memorable.

Emotional Reflection:

“How do you feel about your work in this unit? Do you feel proud or embarrassed?”

One thing I will always be is accountable, even if it brings me down in some manner. Not only do I grow but I never get caught in lies because I stick to the truth. I can honestly say that I do not have one piece of work that I feel I put 100% of my effort into;like I normally do. I am willing to admit that it just wasn’t my priority and I felt like I wasn’t my normal, academic self. No matter what, I will complete work to the best of my abilities. During the unit, I struggled to adapt to the new curriculum (as a lot of folks did, I'm sure,) I let myself fall behind in some moments because I felt the need to complete other things. This reflection sounds very self-critical, which it is, but even though I may have fallen behind and stressed too much, I know I can always,always do a better job!

Over-All:

“What Does the Future Look Like?”

This unit in math has probably been the most difficult to adjust to. I have always been very smart and able to figure things out, especially when challenged. Though, I find that math has always been easier to learn in person and on paper. I value the use of old fashioned work when it comes to math. I think what made me less interested in math this year was the fact that I had too much to do and it was online. Going into the next unit of math, I will make sure that I can use the resources that I need in order for me to be passionate again.

“How did the unit improve your understanding of the relationship between algebra and geometry?”

The transition from algebraic math to geometric can be difficult, especially when applying new and old methods. It's important to understand each of the subjects and to review them before understanding pre-calculus methods and beyond; they both create a base for learning.For me, personally, I see algebra as a way to dig deep into graphs, formulas; Geometry seems to be more about shapes and understanding space and angles within them. Even if that is not what they are truly about, to me I find that the best way to categorize them. During the unit, we were mainly focused on real proof, not just the problem and answer. Learning how to fully prove something is a good way to understand the relationship between these two concepts. Once I was able to fully understand and actually prove a concept during the unit, it made it more valuable and memorable.

Emotional Reflection:

“How do you feel about your work in this unit? Do you feel proud or embarrassed?”

One thing I will always be is accountable, even if it brings me down in some manner. Not only do I grow but I never get caught in lies because I stick to the truth. I can honestly say that I do not have one piece of work that I feel I put 100% of my effort into;like I normally do. I am willing to admit that it just wasn’t my priority and I felt like I wasn’t my normal, academic self. No matter what, I will complete work to the best of my abilities. During the unit, I struggled to adapt to the new curriculum (as a lot of folks did, I'm sure,) I let myself fall behind in some moments because I felt the need to complete other things. This reflection sounds very self-critical, which it is, but even though I may have fallen behind and stressed too much, I know I can always,always do a better job!

Over-All:

“What Does the Future Look Like?”

This unit in math has probably been the most difficult to adjust to. I have always been very smart and able to figure things out, especially when challenged. Though, I find that math has always been easier to learn in person and on paper. I value the use of old fashioned work when it comes to math. I think what made me less interested in math this year was the fact that I had too much to do and it was online. Going into the next unit of math, I will make sure that I can use the resources that I need in order for me to be passionate again.