Unit+1

Unit 1, Lesson 1 What are the similarities and differences between a natural number, whole number, and integers? The similarities of a natural number, whole number, and integers are that they all include positive numbers and have no fractions. A difference between them all is that natural numbers only include positive numbers, so not including 0, while the whole numbers and integers do have 0. But whole numbers only include 0 and positive numbers. Integers include negative numbers along with positive numbers and 0.

What is the difference between a rational and irrational number? The difference between a rational and irrational numbers is that irrational numbers have non-terminating and non-repeating decimals. Rational numbers don't, and they have terminating or repeating decimals.

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. This statement is false. A reciprocal of a positive real number doesn't have to be less than 1. A positive real number includes 1, and the reciprocal of that number, 1/1, is equal to 1, not less than 1.

True or False: An integer is a rational number. Explain your answer and use an example if necessary. True. An integer includes positive numbers, negative numbers, and 0, but no fractions. Rational numbers are fractions and includes positive and negative numbers and 0 as well. Everything that belongs in the integer category also belongs in the rational number category because the rational number can cover everything in that category and more. The positive and negative numbers (that are whole numbers) and 0 can be represented as a fraction. Ex. 3 (integer) --> 3/1

True or False: A rational number is an integer. Explain your answer and use an example if necessary. False. Rational numbers have fractions such as 3/7. Integers don't and only have negative numbers, positive numbers, and 0, so no all rational numbers are integers. "A rational number is an integer" isn't always true.

True or False: A number is either rational or irrational, but not both. Explain your answer and use an example if necessary. True. Rational numbers have fractions that in another form, are terminating or repeating decimals. For irrational numbers, they are any non-repeating, non-terminating decimals. A number can't be non-repeating, non terminating, but also terminating and repeating at the same time. They are opposites.

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) 3/4, .75 A rational number that is infinitely repeating (represented in both fraction and decimal form) 1/3, .33 (with a dash on top of the threes) A real number that fits at least 4 categories of the real number system and explain verbally how that number fits in each category 1 --> A natural number because it is a positive number only and not a fraction, a whole number because it is either 0 or a positive number (it is a positive number and not a fraction), an integer because out of being a negative number, a positive number, and 0, it is a positive number and not a fraction, and it is also an rational number because it can be represented as a fraction, 1/1.

Unit 1 Lesson 2 What is the difference from using brackets [] and parenthesis in interval notation. How does this notation relate to graphing an inequality? Using brackets mean that the interval is greater than/less than and equal to than a certain number, but using parentheses mean that the interval is greater/less than a certain number, and not including that certain number. It relates to graphing an inequality because it can represent what the inequality and show its interval.

What is the difference between a bounded and unbounded interval? Bounded interval means that the interval is between two numbers (or includes those two numbers). Unbounded interval means that the interval starts at a point and continues into positive or negative infinity.

What is the reasoning for only using parenthesis when infinity is included in your interval? Parentheses are used because it shows that could continue forever. There is no end/starting point to infinity, so as when using interval notation, it should be open so it doesn't include any end/start.

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. ( - infinity, 10), x < 10, x is any real number less than 10 [4, 11), 4 < or equal to x < 11, x is any real number between 4 and 11, including 4 and not including 11

Unit 1, 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. Three methods that can be used to find the distance between two points is by graphing or counting, using the distance formula, or applying the Pythagorean Theorem. Graphing and counting would be used when the points lie along the same vertical line or same horizontal line. Using the distance formula is convenient when there are no graphs given and the points don’t have the same y axis or x axis. And applying the Pythagorean Theorem is convenient when one point lies on a horizontal line and the other lies on the vertical line on a graph.

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. Three methods that can be used to find the midpoint of two points on a coordinate plane are by counting, using the midpoint formula, or by using the graph. The first method is convenient to use when the points are on a vertical or horizontal axis on a graph. The midpoint formula is fitting when there is no graph given and none of the points have the same x or y coordinate (they are diagonal). And using the graph to draw the intervals is convenient when you’ve forgotten the midpoint formula and the points are on a graph, but are displayed diagonally.

Given the link to the following example, explain in your own words what is going on during each step of the problem.


 * 1) The verbal representation is put into an algebraic representation.
 * 2) The coordinates given are used as the first set of coordinates. They the coordinate (0, y) is used as the second set of coordinates, where y is the unknown number on the y axis. The distance is 5. (Labeling the variables)
 * 3) Writes the algebraic representation.
 * 4) Fills in the variables with the labels.
 * 5) Simplifying the numbers on the right side under the radical sign (such that (0 – 4) ^2 is 16)
 * 6) Expanding the term (y + 4)^2 into a trinomial
 * 7) Simplifying the expression under the radical sign (like terms go with like terms, so 16 + 16)
 * 8) Square each side of the equation to get rid of the radical sign
 * 9) The equation shows the result after squaring (so 5^2 would be 25 and the other side would be the same expression but there is no radical sign over it)
 * 10) Move 25 over to the other side so that it can be solved (quadratic equation style) The equation needs to equal 0.
 * 11) Factor the trinomial on one side
 * 12) Solve, so the answers should be y = -1, and y = -7
 * 13) Put in coordinate form. The answers are y coordinates.

Unit 1, Lesson 5 What is the standard form equation of a circle with a radius of (0, 0) Standard form equation would be 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. To find the center of a circle given two endpoints of the diameter, use the midpoint formula and find the midpoint of the two points. That is the center point.  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 using the distance formula to find the distance between those two points, and the result would be the length of any radius in 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 tomorrow’s class discussion. To be tangent means that a line touches the curve at only one point. ( []) à  For a circle to be tangent to the x-axis, I predict that the x-axis only shares one point with the circle. And for a circle to be tangent to the y-axis, I predict that the y-axis only shares only one point with the circle.