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. 

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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.