Unit+1+Virtual+Notebook

Lesson 1


 * **What are the similarities and differences between a natural number, whole number, and integers?**
 * A natural number is all positive numbers not including zero. Whole numbers are natural numbers including zero. Integers are positive and negative numbers that include zero. Neither of these three types of numbers include fractions.
 * **What is the difference between a rational and irrational number?**
 * A rational number can be written as a terminating or repeating fraction, while an irrational number is a non-repeating, non-terminating number.
 * ** Explain if the reciprocal of a positive real number must be less then one. If this statement if false prove your argument with an example and explanation. **
 * No; the reciprocal of a positive real number may be less than one. For example, take any number from zero to one, such as one half. One half is a positive real number, and the reciprocal of it, two, is greater than one. Thus, this statement is false.
 * ** True or False: An integer is a rational number. Explain your answer and use an example if necessary. **
 * True; an integer is a positive or negative whole number that includes zero, so they can be written as terminating fractions. For example, -2 is an integer, and written as a terminating fraction it would be -2/1.
 * ** True or False: A rational number is an integer. Explain your answer and use an example if necessary. **
 * Sometimes true. It depends on which rational number it is; for example, 2 is a rational number, and is obviously an integer. However, 4.25 is a rational number since it terminates, but is not an integer since it is not a whole number.
 * ** True or False: A number is either rational or irrational, but not both. Explain your answer and use an example if necessary. **
 * True. If a number is rational, it means it terminates or repeats. That contradicts the definition of an irrational number, which doesn't terminate or repeat. Therefore, this statement is true.
 * **Give an example of a real number set that includes the following elements:**
 * ** A rational number that is terminating (represented in both fraction and decimal form) **
 * 2.5, or 5/2.
 * ** A rational number that is infinitely repeating (represented in both fraction and decimal form)**
 * 3.333..., or 10/3.
 * ** A real number that fits at least 4 categories of the real number system and explain verbally how that number fits in each category **
 * 6; this number is a natural number, since it is a positive whole number; it is a whole number for the same reason; it is an integer because it is either a positive or negative whole number; and it is a rational number because it can be expressed as a terminating fraction (6/1).
 * ** A rational number that is terminating (represented in both fraction and decimal form) **
 * 2.5, or 5/2.
 * ** A rational number that is infinitely repeating (represented in both fraction and decimal form)**
 * 3.333..., or 10/3.
 * ** A real number that fits at least 4 categories of the real number system and explain verbally how that number fits in each category **
 * 6; this number is a natural number, since it is a positive whole number; it is a whole number for the same reason; it is an integer because it is either a positive or negative whole number; and it is a rational number because it can be expressed as a terminating fraction (6/1).
 * ** A real number that fits at least 4 categories of the real number system and explain verbally how that number fits in each category **
 * 6; this number is a natural number, since it is a positive whole number; it is a whole number for the same reason; it is an integer because it is either a positive or negative whole number; and it is a rational number because it can be expressed as a terminating fraction (6/1).
 * 6; this number is a natural number, since it is a positive whole number; it is a whole number for the same reason; it is an integer because it is either a positive or negative whole number; and it is a rational number because it can be expressed as a terminating fraction (6/1).

Lesson 2


 * **What is the difference from using brackets [] and parenthesis in interval notation. How does this notation relate to graphing an inequality?**
 * Brackets in interval notation signify that a number is greater than or equal to or less than or equal to. Parentheses mean that a number is either greater than or less than. It relates to graphing an inequality because a bracket shows that a point must be filled in and the parentheses shows that a point must be left blank.
 * **What is the difference between a bounded and unbounded interval?**
 * A bounded interval has two endpoints, meaning that it is surrounded by two actual number values (not infinities). An unbounded interval has one endpoint left open; in other words, one of the endpoints is an infinity.
 * **What is the reasoning for only using parenthesis when infinity is included in your interval?**
 * A parentheses is used for an infinity because that's what an infinity is; never ending. We can't use brackets because nothing can be equal to an infinity; it's not an actual number.
 * **Give an example of a bounded interval and an unbounded interval. Represent the interval as an inequality and verbal. You may not use an example shown in your reading.**
 * A bounded interval would be something like (8,66), or x is greater than 8 but less than 66. An unbounded interval would be (-** ∞ **,6), or x is less than 6.
 * **Give an example of a bounded interval and an unbounded interval. Represent the interval as an inequality and verbal. You may not use an example shown in your reading.**
 * A bounded interval would be something like (8,66), or x is greater than 8 but less than 66. An unbounded interval would be (-** ∞ **,6), or x is less than 6.
 * A bounded interval would be something like (8,66), or x is greater than 8 but less than 66. An unbounded interval would be (-** ∞ **,6), or x is less than 6.

Lesson 3


 * **What are three methods you can use to find the distance between two points on the coordinate plane? Explain when it is most convenient to use each method.**
 * To find the distance between two points, you can count, use the distance formula (sqrt(x2-x1)^2 + (y2-y1)^2)), or use the Pythagorean theorem. It is most convenient to use counting when the two points have the same x or y value. The distance formula can be used in any situation but it can be quite lengthy. The Pythagorean theorem is most conveniently used when two points are diagonal from each other; form a right triangle and use the theorem.
 * **What are three methods you can use to find the midpoint of two points on a coordinate plane? Explain when it is most convenient to use each method.**
 * To find the midpoint of two points, you can find the median of the two x or y values and use that median with the corresponding x or y value in order to form the midpoint. Finding the median of the two x or y values only works if the other y or x values are the same, respectively. So this method won't work if the segment is oriented diagonally. You can also find the whole length of the segment, and cut it in half, and move that many spaces from one endpoint to the other. This method isn't as reliable if segments aren't straight lines on the coordinate plane. The last and most fool-proof method is using the midpoint formula( (x2-x1)/2, (y2-y1)/2 ). This can be used in any situation and yield a 100% correct answer.
 * **Given the link to the following example, explain in your own words what is going on during each step of the problem.**
 * 1. Distance Formula
 * 2. Decide which values correspond to which part of formula
 * 3. Distance Formula
 * 4. Substitute values into distance formula
 * 5. Simplify (0-4)^2
 * 6. Expand (y+4)^2
 * 7. Add the like terms (16+16)
 * 8. Square both sides to eliminate radical
 * 9. Evaluate square of both sides
 * 10. Subtract both sides by 25 to make left side equal to 0
 * 11. Factor trinomial
 * 12. Definition of expression equaling to 0 by zero product property; one of the y values must add up to equal zero, so y= -1, -7
 * 13. Substitute y for -1 and -7, write resulting coordinate points
 * 12. Definition of expression equaling to 0 by zero product property; one of the y values must add up to equal zero, so y= -1, -7
 * 13. Substitute y for -1 and -7, write resulting coordinate points

Lesson 5


 * **What is the standard form equation of a circle with a radius of (0, 0)**
 * x^2 + y^2 = r^2
 * **Explain in words how you can find the center of a circle if you are given the two endpoints of the diameter.**
 * Given the endpoints of the diameter, you can find the center of a circle. Find the midpoint of the diameter, which will give you the center of your circle, since the diameter must pass through the center of the circle.
 * **Explain in your own words how you can find the radius of a circle if you are given the center and a point on the circle.**
 * You can find the radius of a circle given the center and a point on the circle by finding the distance between those two points, since the radius is the distance between the center and a point on the circle.
 * **Using another resource, write the mathematical definition of the word tangent in your own words (remember to include the name of the resource you used). Predict what you think it means for a circle to be tangent to the x-axis? Predict what you think it means for a circle to be tangent to the y-axis? You may change your predictions after tomorrows class discussion.**
 * Tangent: (courtesy of http://www.thefreedictionary.com/tangent) touching at a single point.
 * If a circle is tangent to the x or y axis, I think that it means that it is touching either the x or the y at a single point.
 * **Using another resource, write the mathematical definition of the word tangent in your own words (remember to include the name of the resource you used). Predict what you think it means for a circle to be tangent to the x-axis? Predict what you think it means for a circle to be tangent to the y-axis? You may change your predictions after tomorrows class discussion.**
 * Tangent: (courtesy of http://www.thefreedictionary.com/tangent) touching at a single point.
 * If a circle is tangent to the x or y axis, I think that it means that it is touching either the x or the y at a single point.