Lesson 1
Pizza - Central Angle, Sectors, Arc Length
![Picture](/uploads/3/8/6/0/38601155/8921349.jpg?182)
http://cliparts.co/cartoon-pizza-pictures
A circle's properties can be seen everywhere in our world because many of our systems have been based of the simple circle. A central angle can be seen throughout every pizza slice. If you measure the angle of the tip to the crust, the central angle would be congruent with the arc(crust). An area of a pizza slice can be also a sector which is also another circle property. The arc length can be found for a pizza by first knowing how many equal slices are there. Than you divide 360 by that number. Than you find the total circumference pi*d by finding the diameter. If you apply all the properties and formulas to a pizza, you can find anything of that pizza and all its dimensions. In addition, the simple and known concepts like the radius, the diameter, and chords, circumference can be spotted physically on a pizza. To find the sector area or area of one equal slice of pizza, we need to know the radius. Using pi*r^2 you know the whole area of the pizza. Now to find the area of one slice, you multiply that area by the angle of that one slice of pizza which would be 8 slices hence the angle in this example. So pi*r^2/8 would be the area of one slice or a sector in geometry. This pizza illustration can also be related to the whole unit is it focuses on the concept of circles deeply so you can apply anything learned in the lessons to this everyday object. For example, to cut down on box material for pizza boxes, the boxes must be designed as the edges o the box are tangent and touching the crust of the pizza at 4 points to not waster material. If the box is tangent and touches the crust of the pizza at 4 points, then it will be not wasting as much material. That's why each size of pizza is given a different box instead of the biggest size.
A circle's properties can be seen everywhere in our world because many of our systems have been based of the simple circle. A central angle can be seen throughout every pizza slice. If you measure the angle of the tip to the crust, the central angle would be congruent with the arc(crust). An area of a pizza slice can be also a sector which is also another circle property. The arc length can be found for a pizza by first knowing how many equal slices are there. Than you divide 360 by that number. Than you find the total circumference pi*d by finding the diameter. If you apply all the properties and formulas to a pizza, you can find anything of that pizza and all its dimensions. In addition, the simple and known concepts like the radius, the diameter, and chords, circumference can be spotted physically on a pizza. To find the sector area or area of one equal slice of pizza, we need to know the radius. Using pi*r^2 you know the whole area of the pizza. Now to find the area of one slice, you multiply that area by the angle of that one slice of pizza which would be 8 slices hence the angle in this example. So pi*r^2/8 would be the area of one slice or a sector in geometry. This pizza illustration can also be related to the whole unit is it focuses on the concept of circles deeply so you can apply anything learned in the lessons to this everyday object. For example, to cut down on box material for pizza boxes, the boxes must be designed as the edges o the box are tangent and touching the crust of the pizza at 4 points to not waster material. If the box is tangent and touches the crust of the pizza at 4 points, then it will be not wasting as much material. That's why each size of pizza is given a different box instead of the biggest size.
|
Lesson 2
Everyday Clock - Radians vs Degrees
![Picture](/uploads/3/8/6/0/38601155/77224.gif?177)
http://etc.usf.edu/clipart/48400/48481/48481_nclockb.htm
This lesson was the comparison between radians and degrees and understanding the basics of radians as well as converting from radians to degrees. In the real world, many objects can be used to display radians and degrees. The in-depth concept of radians can be written the the form of a clock with certain fractions. For now, we know 180 degrees is pi in radians and a full circle(360) is 2pi. So if you put this on a clock, yo would put those 2 values opposite each other like pi would be where the number 6 would be and 2pi would be where the number 12 is on the clock. Using this, you can see how the math concept of radians can be related to the object of a clock. When engineers build clocks, they need the make the distance between each number equal and some may use radians and /or degrees to be as precise as possible. In degrees, you can divide 360/12 to find the angle measure between each number and the same process goes for radians.
This lesson was the comparison between radians and degrees and understanding the basics of radians as well as converting from radians to degrees. In the real world, many objects can be used to display radians and degrees. The in-depth concept of radians can be written the the form of a clock with certain fractions. For now, we know 180 degrees is pi in radians and a full circle(360) is 2pi. So if you put this on a clock, yo would put those 2 values opposite each other like pi would be where the number 6 would be and 2pi would be where the number 12 is on the clock. Using this, you can see how the math concept of radians can be related to the object of a clock. When engineers build clocks, they need the make the distance between each number equal and some may use radians and /or degrees to be as precise as possible. In degrees, you can divide 360/12 to find the angle measure between each number and the same process goes for radians.
Lesson 3
Bicycle Chain/Conveyor Belt - Properties of Tangents
http://s3.amazonaws.com/illustrativemathematics/images/000/000/575/large/wheelbeltsol_dad9c9a8b7b125f862f77401440c650a.jpg?1332223394
http://i.stack.imgur.com/qvV2j.png Tangents are a major part of mathematics and you can find them anywhere in the real world. One friendly and respected example would be the classic and ancient bicycle chain. Tangents can be found around the two circles and their connection to the axle of the chain moving the bicycle itself. Engineers use the property of tangent when building the chains and making their precise dimensions. If the chain is perfectly tangent to the chain axle, the the chain will run smoothly between both circular axles making the bike also move smoothly. These diagrams can also be looked at as conveyor belts which work on the same system of mechanics. To transport heavy objects, you need to have the belt tangent to the two ends of the system which is the wheel that moves the belt. If it's tangent, the belt like the bicycle chain will work more smoothly and successfully. But tangents can be found in literally every circular object in our everyday lives. These are called tangent pulley systems and can be found in almost all complex mechanical objects like cars. On the bottom is the famous Dom Luis Bridge in Portugal. Engineers used tangent to make the supporting structure of the bridge stronger because of the tangent design and the underlying arc(part of a circle). |
Lesson 4
Bicycle Wheel - Properties of Chords
http://george.karaminas.name/ebay/lm_rad_os/images/lm_rad_os_03.jpg
The tennis racket has numerous chords. If you imagine the head of a tennis racket a circle, you can then find numerous chords. Engineers use chords to build the strength of the tennis racket when hitting tennis balls. A special design to generate more power towards the middle is created throughout eh numerous chords throughout the racket. Although only 1 string is used, putting the string in the concept or idea of chords can be used for many reasons. Chords start from the ends of the racket and those chords are the shortest and they get longer and longer as the string goes to the middle of the racket center or sweet spot. That is the reason why when the ball hits the middle of the tennis racket and has correct contact, you will have the most control and power - because of the chords. |
Lesson 7
Giant Wheel - Angles Created Inside and Outside a Circle
![Picture](/uploads/3/8/6/0/38601155/372682.jpg?250)
http://highered.mheducation.com/sites/dl/free/0070973407/645442/ML9_ch10_quiz_Q10.jpg
This lesson was about the relations of angles everywhere, inside and outside of a circle. We are introduced to simple formulas to find missing values like the inscribed angle theorem in the previous lesson which is also applied to this concept. The carnival's classic giant wheel is the perfect and relevant example that utilizes all these ideas into one real-life instance. As you can see through the outlining of the giant wheel's base and structure, geometry and angles are well-integrated throughout. The exterior angle which outlines like a triangle and is the base and the axle of the wheel, is one of our theorems. The exterior angle is half of the difference of the outer arc and inner arc which is intercepted by the segments. The tangent line is also present in the giant wheel. It is the ground in the real world. The angle located on the circle itself is a theorem as well. E is half the measure of the chords' intercepted arc measure. W can also use the inscribed angle theorem and intercepted arcs to find missing values as well. As you can see, all of these details in the giant wheel are relative in unit 8 and specifically lesson 7's theorems and corollaries.
This lesson was about the relations of angles everywhere, inside and outside of a circle. We are introduced to simple formulas to find missing values like the inscribed angle theorem in the previous lesson which is also applied to this concept. The carnival's classic giant wheel is the perfect and relevant example that utilizes all these ideas into one real-life instance. As you can see through the outlining of the giant wheel's base and structure, geometry and angles are well-integrated throughout. The exterior angle which outlines like a triangle and is the base and the axle of the wheel, is one of our theorems. The exterior angle is half of the difference of the outer arc and inner arc which is intercepted by the segments. The tangent line is also present in the giant wheel. It is the ground in the real world. The angle located on the circle itself is a theorem as well. E is half the measure of the chords' intercepted arc measure. W can also use the inscribed angle theorem and intercepted arcs to find missing values as well. As you can see, all of these details in the giant wheel are relative in unit 8 and specifically lesson 7's theorems and corollaries.