Design a Spinner

1. How did you decide how to design your spinner?

(The information provided above is written in a way called probability notation. Basically what P(red)=40% means it that the probability of the spinner landing on the red section is 40%.)

First: So what I did first was understand and decode the instructions that I already had...
1. Since 40% is just a fancy way of saying 4/10 or 40/100, I put four down on the row marked with the red square to the right (by using the blue up arrows). This then made the size of red "4" or "40%".

2. Then, since 20% is just a fancy way of saying 2/10 or 20/100, I put 2 down on the row marked with the blue square to the right (by using the blue up arrows). This then made the size of blue "2" or "20%".

3. Finally, I repeated the same process as I did with the red column (same logic and exact same actions) because in the criteria above the probability of both red and blue needed to be 40%.

The end result of sizing the colors looked like this...


Next step: I needed to figure out what the hard probability notations P(red or odd number)=100% and P(blue and 3)=20% meant. P(red or odd number)=100%... What I figured out from this is that all three parts needed to be odd numbers to ensure that there was at least a 100% chance that the spinner would land on either red or an odd number. P(blue and 3)=20%... I realized that because blue already took up 20% of the spinner, all I really had to do was label the section with the number "3". So by naming just random odd numbers off the top of my head, I decided to label red "1" and white "5":

1. I labeled the red section "1" as it is an odd number which supports the probability notation of P(red or odd number)=100%.

2. I labeled the blue section "3" as it is an odd number which supports the probability notation of P(blue and 3)=20%. I explain above that blue already takes up 20% of the spinner so the 20% would be labeled "3".

3. I labeled the white section "5" as it is an odd number which supports the probability notation of P(red or odd number)=100%.

In the end my final spinner looked like this...

2. How did you confirm that your spinner design met the criteria?
The criteria to be met:


I already know that the probability of the spinner landing on red and white is 40% each because they each take up 2/5 of the space on the face of the spinner. Therefore, the white and red sections of the spinner will have a higher chance of occurring than the blue section. However, this is what I expected, it is an inference that I inferred, but this may not necessarily happen. The improbable occurence would be if the spinner landed on the blye section more times than the other parts. This is improbable but not impossible. Secondly, for the probability of red or an odd number to be 100% landed on by the spinner, I realized that all the numbers on the spinner have to be odd in order for this to be a true statement, which they are. For the last parameter, I realized that blue is taking up 20% of the space, so all I had to do was label this section with the number "3"/ All of this shows that my spinner successfully meets all the criteria  required for this assignment. To test my assumptions and inferences, I ran an experiment, spinning the spinner 10,000 times:

My experiment results showed that the red and white sections were landed on at almost the same amount of times. It also shows that the blue section was landed on about half the amount of times the red and white landed on. 

3. Is there more than one acceptable spinner design?  If so, what will the spinners have in common? What will be different?
Yes, of course, mostly everything in the world has multiple solutions to it. As I said earlier on in this post, the number "3" as labeled on the blue section must stay blue in order to meet the criteria explained above. However, the other two number can be any odd number as the criteria doesn't specifically call upon certain number. For example: 
 
You could say that there will be an infinite amount of spinner as there are an infinite amount of odd numbers. But this is the only factor of the spinner's criteria that can vary. Otherwise, everything else stays the same and the proportions or sizes of the colors don't change and neither does anything else.

What Do You Expect Mathematical Reflections 1

1. How do you find the experimental probability that a particular result will occur? Why is it called the experimental probability?
          How: To find experimental probability you need to first perform the experiment at hand. Whether it's tossing coins or cups, flipping spoons, or even spinner a spinner, you have to first perform the experiment. For example, lets say that Mary wants to decide which desert to eat for dinner everyday of June-- either fruit, heads, or ice cream, tails. She does a trail to see what the probable outcome will be, tossing the coin 30 times. She needs to record the number of heads out of 30 and the number of tails out of 30. You would perform this experiment and record the results-- heads: 14 times and tails: 16 times. Secondly, you would right out the fraction/relative frequency using formal probability notation with the certain parameters that were chosen at the beginning of the experiment-- heads: 14/30 and tails: 16/30. Then, you would convert that fraction into a percentage, and after this, you can finally begin analyzing the data and begin making predictions about future outcomes.
          Why: The experimental probability, relative frequency, or also known as the empirical probability, is the ratio of the number of outcomes in which a specified event occurs to the total number of trials in an experiment. Lastly, it's called experimental probability because the predictions and the outcomes are based off trials formed from an experiment.

2. In an experiment, are 30 trials as good as 500 trials to predict the chances of a result? Explain.
          NO... NEVER! In an experiment 30 trials is not as food as 500 trials to predict the chances of a result. The golden rule is that the more data, trails, and evidence you have, the better. When you are performing an experiment, you have to always try to have somewhat accurate results otherwise this doesn't help you in getting the information needed to solve your question or make further choices and predictions. Again, the more trials you have, the more accurate the results of you experiment will be as you have more and more data to work with. To predict the chances of a certain result through the experiment, you need to base that prediction off of good data that's somewhat reliable and can be almost accurate. A connection to a previous unit is the Are You Normal unit, the math teachers collect a huge sample size of all math students in a survey and then we analyzed the data. To do this experiment in the previous unit we need a big sample as the it was a very large survey and there weren't that many common answers. So to make a prediction on the population we needed a large sample, otherwise the data would be very hard to analyze. This idea of 500 trials being better than 30 trials directly relates to this previous unit.

3. What does it mean for the results to be equally likely?
          For results to be equally likely, they all have to have the same chances of occurring. It also means that the experiment has to be unbiased and fair. However, equally likely doesn't mean that results have a 50% and 50% chance. The most important part of determining if something is equally likely to another, you have to now how many parameters were set for the experiment. For example, if you have 4 different parameters or options, you would have to distribute these parameters or 100% evenly among these 4 parameters. Therefore, they would have the same chance of occurring, making none of them less likely to occur than another. Going with the example below each parameter or result would have  25% chance of occurring, and the results would be equally likely.

UN Day- Blog Reflection

1. Give a brief description of UN Day and the activities you undertook.

RLA: In RLA we had these two amazing poets come in and share with us some of their knowledge. They taught us quite a lot about themselves as writers and taught us some tips and advice when writing poetry. Then they taught us about P.I.P.E.S. which stands for projection, inflection, practice, eye contact, and stance. The next day we had a large poetry reading in the library. In this poetry reading we got to put our names into a ballot box where the poets would call out our names and then we would be able to recite one piece of poetry that we wrote to all our peers, teachers, and our guest poets.

Math: In math we did a lot of logical and debatable reasoning and discussing. So one thing that we did was playing the role of hospital in a country where liver transplants were needed and were very rare to get, so we had to pick out of 10 people which 4 we would save. To decide which people to save we had to think of the important factors such as age, importance, role, money, etc. The next thing we did was model people in the UN who represent different countries. Based on our countries per family monthly income we would debate over food, shelter, and medical care. The country that debated for the most money would win the basic needs debated on, but we soon came to the decision that this was unfair. It was unfair because some countries weren’t as fortunate so they weren’t getting anything, so then we came up with an agreement that suited most of the countries.

PE: With a partner or alone we carried different sizes of buckets around the track for two laps. This was to stimulate how people in poor countries who didn’t have access to clean water nearby had to walk long and tiring journeys just so that they would be able to survive. They needed to take 10 mile walks at the most just to get something that we all take for granted- water. This was a very tiring experience and it made my arm, back, and legs hurt. It was also very hard to walk without spilling any water under the hot sun.

Science: In science we the most interesting game which actually had a lot of meaning hidden behind it. In this game we were split up by sides and needed to get a source of food, water, and medical aid. We had a box that needed to be filled with rice grains to symbolize food, another box that needed to be filled with water to symbolize water, and a box that needed to be filled with 20 ribbon to symbolize medical aid. We had to in pairs had to run into science rooms to get these supplies and the quickest team to get them won. But what we realized was that the game was also unfair. A side was closer than any side to all the facilities, the teachers were helping them, and they had bigger spoons and cups to get water and food with. This conveyed what different countries were going through and how sometimes things can be unjust. In developing countries things are harder to get and sometimes resources are limited, but in more developed countries they have more advantages and they can get their basic needs.

Food Festival: All of the students brought in food that was connected with a country that they felt was important to them. Then they brought it to the cafeteria where we had 3 tables of a variety of food. Then for lunch we got to experience different kinds of food from around the world.

2. Choose two questions from the list below to answer thoroughly, using descriptions and detail.

• Did you learn a new skill or discover an interest during UN Day?

A new interest that I have discovered was debating with reasoning and learning about how the UN works. We participated in this activity in Math class and I really enjoyed it. I loved how we got to debate with other students in the class over who to give a liver transplant to so that they could survive. I like understanding about the individuals could and have affected the country and their contributions to the development of the country. I also liked the idea of being a country in the UN and having a certain amount of money and coming up with solutions to fix the unjust. This stimulation was very real because we even learnt that the countries who have seemed to won world war II had more power over voting and the decisions that were made.

• During UN Day, did you hear, smell, or feel anything that surprised you?

I felt a lot of unjust, unfairness, and sorrow during UN Day. When we were stimulating the game in science of getting our basic needs I felt as if we were being cheated. As if something wasn’t right and some countries (sides) were being treated differently because of different circumstances. I also felt really fortunate and sad for the people who carry these huge buckets of water for long distances underneath the hot sun. This taught me not to take water and my life for granted. Also in math I felt kind of bad for the groups who were poor countries and didn’t have a lot of money. They really didn’t have any choice and they couldn’t do anything about their situation. This got me to thinking about how the UN has a very important purpose to make things as fair as possible.

• How was your UN Day experience different from what you expected?

I thought that the UN day was like what we did in elementary school. I thought that we would learn about all the different cultures in the world and do activities based on those cultures. But actually this experience was actually much funner and more engaging. It taught us something different about the world and gave us a new perspective on life. This was a very engaging day as it was something new. Most of us who have been at SAS for a long time are used to the normal UN day but this was different and funner. I also really liked how the activities that we did were not only entertaining but also taught us a lot through them.  

• What do you view differently now because of your experiences during UN Day?

I view the difference between developing, almost developed, and developed countries differently and how important the UN is. I already knew that there was a difference between the amount of money the countries made and the difference between the qualities of life. I already know that the UN helps countries with their issues and solves problems. But now I realize that the countries maybe just needed some help, maybe they felt as if they weren’t being treated fairly. Maybe they just needed that little boost that advantage. If that wasn’t possible then maybe they just needed an agreement something that sets them the same as other countries something that makes them equal. And that’s why they have the UN, to solve problems, come up with solutions, make things fair and right.

3. Choose 1 question from the list below to answer thoroughly, using descriptions and detail.

• If you could do UN Day again, what would you do the same or differently?

If I were to change anything about UN day I would change the RLA and PE activities. I’m not sure if the RLA activity was because of the authors or if it had to do with UN day for a reason. Maybe we could have a more related activity which helps teach us about the UN and the countries around the world. Maybe the activity was connected through public speaking in the UN, I’m not sure, so maybe they could have explained that. Also in PE some people were spilling 3/4 of the water out of their buckets so that the bucket would be lighter and easier for them to run around the track with twice. I thought that this was very bad and it just defeated the whole purpose of doing the activity. If the people weren’t giving it their all then why even bother, they wouldn’t have really understood what the less fortunate people were going through.

Looking for Pythagoras- Math Reflection Questions Investigation 4

1. Give three examples of fractions with decimal representations that terminate:
A terminating decimal is a decimal that has digits that doesn't go on forever. A terminating decimal is the same thing as a rational number, and doesn't repeat forever. Three examples of fractions with decimal representations that terminate are 1/8,  2/10,  and 1/4. The decimal representation of 1/8 is equal to 0.125, which is terminating because the decimal values don't go on forever. The decimal representation of 2/10 is equal 0.2, which is terminating because the decimal values don't go on forever. The decimal representation of 1/4 is equal to 0.25, which is terminating because the decimal values don't go on forever. 

2. Give three examples of fractions with decimal representations that repeat:
A repeating decimal is when the decimal values of the number goes on forever, or infinitely. Three examples of fractions with decimal representations that repeat are 1/3, 1/11, and 2/9. The decimal representation of 1/3 is 0.3333333...... (3 repeating on forever). This is an example of a repeating decimal because the three goes on forever. The decimal representation of 1/11 is 0.090909...... (09 repeating on forever). This is an example of a repeating decimal because the three goes on forever. The decimal representation of 2/9 is 0.222222...... (2 repeating on forever). This is an example of a repeating decimal because the three goes on forever. Usually, repeating decimals are shown with a bar over the number or numbers that are getting repeated.

3. Give three examples of irrational numbers, including one irrational number greater than 5:
An irrational number is a number that cannot be expressed as a ratio or a fraction. Three examples of irrational numbers, including one that is greater than 5, are √27, √30, and √34. The square root of 27 is approximately (fraction estimate) 5.196. The square root of 30 is approximately 5.477. The square root of 34 is approximately 5.8309. But remember that these square roots have no ends and the answers that I gave were just estimates. 

4. How can you determinate whether you can write a given decimal as a fraction:
You can determine whether you can write a given decimal as a fraction by looking at it. You automatically know that a decimal can be represented as a fraction if it is either terminating or rational. You also know that if something irrational, that it CAN'T be written as a fraction. If a fraction neither repeats nor terminates, you automatically know that it can't be written as a fraction, which means that it is irrational. When looking at the terminating decimal, you can look at the place value in which the last number is in. For example, let's use 0.02. THe 2 is in the hundredths place, so the fraction is 2/100. You could also guess and check to make the decimal a fraction. 

Realistic Fiction Story

Looking for Pythagoras- Math Reflection Quesions Investigation 2

1. Describe how you would find the length of a line segment connecting two dots on dot paper. Be sure to consider horizontal, vertical, and tilted segments. 

          Horizontal: 

          Vertical:

          Titled: 

There are two strategies for this:

1. You could get a ruler and measure the line segment or you could count the units between the two dots that make up the line segment. This may be accurate in some instances, like when the line is perfectly straight and the measurement ends at a whole number, but sometimes this strategy is inaccurate and hard to get the exact answer with.  

2. You could use the "square method." So you would first have to picture that that line segment is one of the four side of a square. Then, you would draw in the other three sides of the square. To do this, you use the slop of the segment or also known as the "rise over run". After creating a square based off of the simple line segment you would find the area of this square. I have already answered how to find the area in my investigation 1 set of question, but as a quick reminder, you you cut  up the shape into smaller triangles or squares and then add their area's up, or you could draw a bigger square around the square and take away all the excesses pieces from the total area of the bigger square. After finding the area of the shape, all you have to do to find the length of the segment (which is one of the sides of the shape) is find the square root of the area of the shape. 

2. 

a) Explain what it means to find the square root of a number. 
          When you find the square root of a number, you are finding the number that when multiplied by itself, creates the number you are find the square root of. For example, suppose we are trying to figure out the square root of 25. Well, we know that 5*5= 25. So 5 is the square root of 25. As another example, suppose a square has the area of 16 units. Since a square has 4 equal sides, an integer of the length of on of the sides of the square multiplied by itself will come to the total area of the square. The length of the sides of the square is the square root of the area of the square. Coming back to our example, we find out that each of the sides of the square are 4 units long. And since there are four sides to the square which each 4 units long, we can add them all up: 4+4+4+4=16. This proves that the length of the sides of the square is 4 each, as the area adds up correctly. aThis brings us to our final conclusion that the square root of 16 is 4. 

b) Explain whether or not a number can have more than one square root. 
          Yes, a number can have more than one square root. Since, the square of a number is when you multiply  a number by itself to get the digit wanted. For example, take the number 36. The obvious square root is 6. I got my answer as when you multiply 6 by 6, you get 36. However, there is more than just one square root for the square root of 36. The second square root for 36 is -6. The logical behind this answer is that when you multiply two negative integers your answer will always positive. So, -6 multiplied by itself is 36 also. So when writing questions on finding the square root of a number you must specify if you want a negative or positive number. In conclusion, a number can have more than one square root. 

3.

a) Explain what it means to find the cube root of a number.

          To find the cube root of a number is similar to finding the square root of a number. In math, to "cube" something means to times it to the power of the three. So to find the the cube root of a number, you multiply the number by itself three times.  For example to find the cube root of 2 you would perform this equation; 2*2*2=8, Therefore, the cube root of 8, is 2/ 2 cubed is 8.

b) Explain whether or not a number can have more than one cube root.

           A number cannot have more than one cube root. The reason I say this is because, you cannot have a negative number multiplied by itself three times and equal a positive number.  For example, the cube root of 8 is 2, but that doesn't mean the cube root of 8 is also negative 2.  Because -2*-2*-2, equals -8.  Therefore, you can only have one cube root per number.  

Looking for Pythagoras- Math Reflection Quesions Investigation 1

1. In the city of Euclid, how does the driving distance from one place to another compare to the flying distance?

          In most cases, the driving distance from one place to another is longer than the flying distance. When driving you have to drive on roads and you can't cut between land to take short cuts. But, when flying you can fly diagonally cutting across land and not following roads. In other cases the driving distance from one place to another compared to the flying distance are the same. This happens when there is one straight route where not matter what flying or driving, there is no difference in distance. Overall, the flying distance can never be more than the driving distance, it's either less than or equal to the driving distance. 

2. Suppose you know the coordinates of two landmarks in Euclid. How can you find the distance between those two landmarks?
          To find the distance between the coordinates of two landmarks in Euclid, you can subtract x1 from x2, and also subtract y1 from y2. Then you take the absolutes value of x1-x2 and y1-y2. Lastly, you add the two absolute values, which gets you to the distance between the coordinates of two landmarks in Euclid. 

3. What are some strategies for finding areas of figures drawn on a grid?

          There are a couple strategies for finding areas of figures drawn on a grid: 

                    1. If it's a regular shape with "full boxes" you can just count the boxes that make up the area of the figure on the grid. 

                    2. It it's an irregular shape, then you can deconstruct the shape into rectangles and triangles. Then calculate the area of each rectangle or triangle. Lastly, add all the areas of the deconstructed shape together and you have your answer.  

                    3. Enclose the figure on the grid in a larger rectangle/square. Then find out how much the area of the unwanted parts of the larger rectangle/square is. Lastly, subtract the areas of the unwanted parts from the area of the larger rectangle/square.