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**Statistics** **Mathematical Notation** One of the most frequent questions raised by students is why all the math? There is no single answer to this question, but hope to show why math is important. There is no single answer to this question, but we will argue and justify whey we need all those complicated looking formulas. Mathematical symbols are used as a convenient way to represent quantities and concepts that are used frequently. It is possible to write every equation out in terms of words, but this can often become confusing. Look at these examples: a. 1+1 can be written as: one plus one b. 1 + 1 – 7/6+3 can be written as: the quantity one plus one minus seven divided by the quantity six plus three. You can see that mathematical symbols are valuable in that they save space and are less confusing than sever sentences. The only problem is that the student has to learn the shorthand notation. Statistical methods are used all over the world. The fact that everyone uses the same symbols makes it possible for statisticians to communicate with each other. Further, the standardization of symbols can help you directly. All of the accumulated wisdom of a century of statistical development is around to help you when you want it to because everyone uses the same mathematical shorthand. This brings up an important point: You will not learn everything that there is to know about statistics from this class. What you will learn is: 1. How to approach a real-world problem with statistics. 2. How to understand statistical formulas and procedures. 3. How to classify and use data appropriately. Mathematics has always been an important part of statistics and there is no other way to approach the subject. Until a better idea comes along, you must bear the cross of mathematical notation. We will start by looking at the most elementary relationships. This involve the operations involved with **__signs.__** Let’s look at their definitions: General Numbers: a general number is either positive, negative or zero. Positive Number: a positive number is greater than zero. Negative Number: a negative number is less than zero. Zero: the number which separates the negative and positive numbers. This sounds like a lot of words, but it will make sense if you think about it in terms of the following diagram: negative numbers positive numbers ß --negative direction positive direction - à . . .-3 -2 -1 0 +1 +2 +3 . . . Figure 1: The General Numbers (above) The picture shows the numbers laid out in a row or scale and zero is the dividing point between the positives and negatives. The words positive and negative have meaning in terms of **__direction__** away from the zero point. To illustrate this, let us take a peep at a hypothetical person’s checking account: John Doe Bank Account Balance forwarded: $200.00 Deduct check #400 – Acme Market $10.00 New Balance $190.00 Deduct check #401 – Jones Shoe Store $5.00 New Balance $185.00 Credit Deposit $30.00 New Balance $215.00 We can also represent this in terms of our diagram:
 * __A. Why Mathematical Notation?__**
 * Argument 1: Shorthand**
 * Argument 2: Communication**
 * Argument 3: That’s The Way It Is.**
 * __B. Relationships+__**
 * 1. Signs**
 * || [[image:http://mulloa.wikispaces.com/site/embedthumbnail/placeholder?w=295&h=50 width="295" height="50"]] ||
 * || [[image:http://mulloa.wikispaces.com/site/embedthumbnail/placeholder?w=295&h=50 width="295" height="50"]] ||

-200 -100 0 +100 +200 (initial state) Figure 2: Initial State of John Doe’s Checking Account. The next two events in Mr. Doe’s checkbook represent **__deductions__** (or subtractions) of 10 and 5 dollars respectively. The operation of subtraction can be graphically represented in terms of movement in the negative direction, as shown in Fig. 3 0 +185 +200 B A Figure 3: Negative direction movement of J. Doe’s bank account.

The next thing that happened to John Doe’s account is that he deposited (or added) 30 dollars. The operation of **__addition__** is represented graphically in terms of movement in the positive direction, as shown in Fig. 4:
 * || [[image:http://mulloa.wikispaces.com/site/embedthumbnail/placeholder?w=304&h=50 width="304" height="50"]] ||
 * || [[image:http://mulloa.wikispaces.com/site/embedthumbnail/placeholder?w=304&h=50 width="304" height="50"]] ||

-35 0 +100 +185 +215

Figure 4: Positive direction movement of J. Doe’s bank account. By now you should have deduced the following rules: 1. Every general number has two parts: a sign and an absolute value. Look at this diagram: sign à + and 2 absolute value The sign can be **__plus__** (+) or **__minus__** (-). Exception : zero (0) has no sign. Examples of General Numbers: +2.5, -1.7, +3, -.001, +1.796 Things that are __not__ General Numbers: +0, 1.2, -0, .02 (Why?) Read #1 for answer. 2. The sign of a number tells you tow things: a. It tells you whether the number is greater than (+) or less than (-) zero. b. If the numbers are being combined, it tells you which direction to move.

3. The **__absolute value__** of number tells you how far away from zero the number is from zero. often, statisticians find it useful to talk about the absolute value of a number. They use this notation: /a/ - to be read the “absolute value of “a.” Examples: /+4/ = 4 /-3/ = 3 /+.0001/ = .0001 Many times, you will find that you will be dividing or multiplying numbers with signs. Here are some simple rules: a. A plus and a plus yield a plus sign number. Examples: (+4 x (2) = +8, +4/+2 = +2 b. A plus and a minus yield a minus sign number. Examples (-3) x (+4) = -12, -16/+4 = -4 c. A minus and a minus yield a plus sign number. Examples (-3) x (-2) = 6, -4/-2 = +2 Examples: -(+4) = (-1) x (+4) = -4, -(-4) = (-1) = x (-4) = +4 This special case helps us subtract negative numbers. Look at this example: Problem: Compute (+1) – (-12) Solution: (+1) – (-12) = **(+1) + (-1) x (-12)** =(+1) + 12= If you remember the three simple rules, you should have no trouble combining signed numbers.
 * __Mathematical Notion__**
 * __Definition:__** The absolute value of a number is just the number without the sign.
 * 4. Rules for Multiplying or Dividing with Signs**
 * SPECIAL NOTATION:** Sometimes the expression (-1) x (a) is written as –(a). The two are equivalent.
 * 13**

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I __ntegrated Math 2: SEMESTER 1__

Module 1: Analyzing Functions 1.1 and 1.2 Pages 6 - 10, #1-12, pages 11, #1-12 Page 12, #a,b,c,& d Pages 13 - 1, 7 #1-10 Page 18, # 1-16 Page 22, #a, b, c, & d

Module 1 and 2: Absolute Value Functions 1.3 and 2.1 Pages 23 - 27, #1 - 6, Page 28, #1-15 Page 30 Lesson Performance Task Question Page 31-33, #1, 2, 3, & 4 Page 33 #1-3 Page 36 #1-8

Module 2.1 Absolute Value Functions, Equations, and Inequalities Pages 37 - 42, #1-6, page 2.2 Solving Absolute Value Equations Read pages 45 - 47, do problems on page 48 #1 - 17

2.3 Solving Absolute Value Inequalities Read pages 51- 55, do problems on page 55, #1 - 17

Module 3 Rational Exponents and Radicals 3.1 and 3.2

Module 4 Adding and Subtracting Polynomials 4.1 and 4.2

Module 4 and 5 Multiplying Polynomials 4.3 and 5.1

Module 5 Multiplying Polynomials 5.2 and 5.3

Module 6 Graphing Quadratic Functions 6.1 and 6.2

Module 6 and 7 Connecting Intercepts, zeros, and Factors 6.3 and 7.1

Module 7 Connecting Intercepts, zeros, and factors 7.2 and 7.3

Module 8 Using factors to solve quadratic equations 8.1 and 8.2

Module 8 and 9 Using square roots to solve quadratic equations 8.3 and 9.1

Module 9 Using square roots to solve quadratic equations 9.2 and 9.3

Module 9 and 11 Quadratic Equations and complex numbers 9.4 and 9.5

Module 11 Quadratic Equations and Complex Numbers

11.1, 11.2, and 11.3

__SEMESTER 2 INTEGRATED MATH 2__

Module 14: Proofs with Lines and Angles 14.1 and 14.2 14.3 and 14.4

Module 15: Proofs with triangles and quadrilaterals 15.1 and 15.2 15.3 and 15.4 15.5, 15.6 and 15.7

Module 16: Similarities and Transformations 16.1 and 16.2 16.3 and 16.4

Module 17: Using Similar Triangles 17.1 and 17.2 17.3 and 17.4

Module 18: Trigonometry with right angles 18.1 and 18.2 18.3 and 18.4 and 18.5

Module 21 Volume Formulas 21.1 and 21.2 21.3 and 21.4 21.5

Module 10 Linear, Exponential and Quadratic Models 10.1 and 10.2 10.3 and 10.4 10.5

Module 22: Introduction to Probability 2.1 and 2.2 2.3 and 2.4

Module 23: Conditional Probability and Independent Events 23.1 and 23.2 and 23.3

__Integrated Math 3 Semester 1__

Module 5: Polynomial Functions 5.1 and 5.2 5.3 and 5.4

Module 6: Polynomials 6.1 and 6.2 6.3, 6.4, and 6.5

Module 7: Polynomial Equations 7.1 and 7.2

Module 8: Rational Functions 8.1 and 8.2

Module 9: Rational Expressions and Equations 9.1, 9.2 and 9.3

Module 10: Rational Functions 10.1, 10.2, and 10.3

Module 11: Radical Expressions and Equations 11.1, 11.2, and 11.3

Module 20: Gathering and Displaying Data 20.1 and 20.2

Module 21: Data Distributions 21.1, 21.2, and 21.3

Module 22: Making Inferences from data 22.1, 22.2 and 22.3

__Integrated Math 3 Semester 2__

Module 12: Sequences and Series 12.1, 12.2 and 12.3

Module 13: Exponential Functions 13.1 and 13.2 13.3 and 13.4

Module 14: Modeling with Exponential and Other Functions 14.1 and 14.2

Module 15: Logarithmic Functions 15.1 and 15.2

Module 16: Log Properties and Exponential Equations 16.1 and 16.2

Module 17: Trigonometry with All Triangles 17.1 and 17.2 and 17.3

Module 18:Unit Circle and Definition of Trig Functions 18.1, 18,2 and 18.3

Module 19: Graphing Trig Functions 19.1 and 19.2 19.3 and 19.4