MA224



=MA224 Mathematics for Elementary and Middle Grade Teachers (Term III)=

=What's New: (UPDATED 2/9/16) =

The Unit 2 Study Guide is posted on the Homework page below. The new Unit 2 Review is posted on the Homework page below. Quiz 2 is posted on the Homework page below.

**Course Syllabus** Course Schedule and Homework **(** 2/9/16 **)**

**Chapter 1 Discussion Notes** **(** 1/12/15 **)** Common Core Standards Companion Website
 * [[file:MA224Ch1.pdf|My Notes]]**

Related Files:

At this point, you should be thinking about how to approach mathematics as a problem solving method. Remember, it's all related. When a high school student is fighting an algebraic expression, at the same time there is a 2nd grader filling in the next number in the same sequence. **Build the bridges that tie this stuff together!**

1. Understand the problem 2. Plan the approach 3. Try the plan 4. Check the result
 * __Problem Solving Method__**

1. Draw a picture 2. Create a model 3. Make a table 4. Work backwards 5. Many more.....
 * __Types of Plans__**

__**Patterns**__ A few more __//Fibonacci//__ occurrences: In the average adult male human hand, the bone at the tip of the middle finger is 2cm, the bone between the first and second knuckle is 3cm, the bone between the second and third knuckle is 5cm, and the bone from the third knuckle to the wrist is 8cm. Furthermore, regardless of the size or gender of the hand, the ratio of bone sizes remains 2:3:5:8. In music, there are 13 notes across the span of an octave. 8 whole notes and 5 sharps/flats. A scale is 8 notes, of which the 3rd and 5th notes are the foundation of the chord which is 3 notes. Also the 3rd and 5th notes are the base harmonizing notes. Lots of Fibonacci numbers floating around out here.

Pascal's Triangle has more to offer as well. Here is a pretty simple site on the [|Pascal Triangle Patterns].

__**Other Conversations (for fun, right?)**__ [|4 Color Theorem] [|Fermat's Last Theorem] [|The Lottery]

Cooperative Learning Files


**Chapter 2 Discussion Notes** **** __ **Sets and Operations** __ The basic set notation is a capitol letter to name the set, followed by an equal sign with a list of number inside a bracket. A = {1, 3, 5,7, 9}. Recall that sets can be empty, and are called **null sets**. There is symbolic notation to refer to an element that belongs to a set. 3∈A means 3 is part of set A.

When comparing sets to one another, we refer to **Unions** and //Intersections// of sets. The notation A ∪ B refers to the union of sets A and B which is a new set containing all the elements in both sets. A ⋂ B refers to the intersection of sets A and B which is a new set containing the overlapping or common elements of both sets. These ideas are easily illustrated with **Venn Diagrams**.

Represents the union of two sets Represents the intersection of two sets

In the Venn Diagram, the outlined box defines the **universe**. The circles represent subsets of the universe. Sometimes the circles overlap, implying that an intersection of elements exists. If the circles fail to overlap, the intersection is considered null. The **complement** of a set would be all of the elements in the universe that are NOT part of the set. If the universe is U = {1, 2, 3, 4, 5} and A = {1, 2, 3} then the complement, written A', would be A' = {4, 5}.

__**Logic and Reasoning**__ Deductive reasoning is based on a series of logical arguments. For example, "if you smash you finger with a hammer it will hurt" represents a **premise**. "John smashed his finger with a hammer" represents a **minor premise** (factual statement). "John hurt himself" is the logical **conclusion** drawn from the premise and the statement of fact. Generally speaking, a properly structured logical conclusion is considered true as long as the original premise is true. There are a few variations of the logical structure: The **inverse** represents a reversal of the major and minor premise. "if you hurt your finger, you hit it with a hammer" is an inverse argument and lacks logical foundation. While true that if your finger hurts, one possibility is that you hit it with a hammer, there are too many other ways to hurt your finger to consider this argument true. The **converse** represents a negation of the original argument. "if you don't smash your finger with a hammer, it won't hurt" also lacks logical foundation. If I drop a bowling ball on my finger, it will hurt; yet the converse seems to suggest that the only way to hurt your finger is with a hammer. The final structure is the **contrapositive**, which is a combination of the inverse and converse. "if your finger doesn't hurt, you didn't hit it with a hammer" makes sense: since hitting it with a hammer would hurt and it doesn't hurt, clearly you didn't hit it with a hammer.

__**Functions and Graphing**__ A **function** is a pairing of elements in one set with elements in another set. The pairing is called **one-to-one**, meaning that the sets are equal in size and each element has a partner. The set of numbers used to create the first half of the pair is called the **domain**. The set used for the second half of the pair is called the **range**. The pairing is called an **ordered pair**. If A = {1, 3, 5,7, 9} and B = {2, 4, 6, 8, 10}, one possible function (pairing) would be the ordered pairs {(1,2) (3,4) (5,6) (7,8) and (9,10)}.

In further study of functions, you can graph the ordered pairs to determine if there is a pattern. The graphing field used is called the **Cartesian Plane**. The plane is divided into four quadrants divided by a horizontal (x) axis and a vertical (y) axis. Where the axes meet is called the **origin**. Movement to the right of and above the origin is considered positive. Movement to the left of and below the origin is negative.

If the graphed ordered pairs are in the form of a straight line, the points are **linear**. Linear functions have equal slope between points, where **slope** is the ratio (undivided fraction) of the differences in y values over the differences in x values. For (1,2) and (3,4), the difference in y values is 2 and the difference in x values is also 2, thus the slope is 2/2 or 1 (in reduced form). Consider two different pairs (3, 4) and (7,8). The y difference is 4, x is also 4, thus slope is 4/4 or 1. If the slope were to remain 1 for every pairing in the function, the function is linear.(not because of the 1, but because they are equal to each other) Not every function is linear! There are quadratic, cubic, polynomial, logarithmic, exponential, and trigonometric functions as well (to mention a few). The only thing they all have in common is a predefined pattern.

**Chapter 3 Discussion Notes**
 * [[file:MA224Ch3.pdf|My Notes]]**

**__Numeration Systems__**
 * Numerals ** are written symbols for numbers. The first numeration systems were for counting. One of the first documented systems was the 1, 2, 3,4, hand system. "Two hands and three" meant 13. Soon a base 10 system was adapted; however it's important to remember that a numeration base can be number of objects. For example, the Babylonian Base 60 system uses 60 objects in the first numeral position. The second position is the 60 2 place and so on... Here is a [[file:Ancient Numerals.pdf|chart ]]of several ancient numeration systems.

__**Operation Algorithms**__ There are several ways to demonstrate each mathematical operation. Addition can be done by carrying numbers, as partial sums, or in Left-to-Right order. Multiplication can be done with a partial products approach, which lends itself well to the development of algebraic skills later on. Subtraction can be taught as three distinct approaches. "Take away", "Comparison", and "Missing Addend" each illustrate the concept of difference. Division can be shown as "sharing" or "measurement", but remember, measurement will eventually lead to the concept of remainder. It is important to understand each approach as it will help you differentiate your instruction.

__**Properties**__ Each operation has a set of properties, or rules, that represent what you are allowed to do with the operation. Associative and Commutative properties allow us to move numbers and re-group them without changing Addition and Multiplication. Closure property defines a characteristic that means that an operation can be completed without changing number systems. For example, even numbers are closed to addition since E + E = E; however the same is not true of odd numbers since O + O = E. Adding two odd numbers fails to result in an odd number, therefore the process is open. Identity properties illustrate operations that have no effect on a number. For example, adding 0 results in no change to the original addend. The same is true of multiplying by 1. Hence, 0 is referred to as the identity of addition and 1 is the identity of multiplication.

Exponents are a way to represent repeated multiplication. We consider 2 5 to mean 2 x 2 x 2 x 2 x 2. In exponent form, the 2 is the base and the 5 is the exponent. Two major properties of exponents involving multiplying and dividing exponents with like base. By definition a m x a n = a m + n and a m ​ ÷ a n = a m - n
 * __Exponents__**

__**Order of Operation**__ To ensure that mathematical operation are always performed the same way, we follow the following Order when completing multi-step problems: Parentheses Exponents Multiplication Division from L to R Addition Subtraction from L to R

**Chapter 4 Discussion Notes** **** **__Number Theory__**
 * Factor s** and **Multiples** are the building blocks of number theory. Remember factors are the numbers that divide another number, and multiples are the numbers you get by multiplying a given number by the whole numbers. We use notation such as 7|21 to mean 7 divides 21. Remember, numbers can be grouped into two categories: **Prime** and **Composite**. Prime numbers have only 1 and itself as factors. A simple way to filter out the prime numbers is the **Sieve of Eratosthenes**. The process of factoring involves breaking a number down into its prime parts. For example, the number 24 can be factored two ways: **Factor pairs** of 24 [ 1 x 24, 2 x12, 3 x 8, 4 x 6]; and **prime factorization** of 24 [ 2 x 2 x 2 x 3 ].

__**Divisibility Properties: Rules to determine divisibility**__ 2: number ends in 0, 2, 4, 6, or 8 3: sum of the number's digits is divisible by 3 4: last two digits of the number are divisible by 4 5: number ends in 0 or 5 6: fulfills rules for 2 and 3 8: last three digits divisible by 8 9: sum of the number's digits is divisible by 9 10: number ends in 0 11: sum of the alternating digits is equal

__**GCF and LCM**__ The **greatest common factor** of a set numbers is the largest number that divides them all. For example 24 and 32 are both divisible by 2 and 4, but 8 divides them both as well. 12 divides 24 and 16 divides 32, but neither 12 or 16 divide both numbers. Therefore 8 is the GCF. The **least common multiple** of a set of numbers is the smallest number that is a multiple of all the numbers. 3 and 5 have infinite multiples. 3: 3, 6, 9, 12..... and 5: 5, 10, 15, 20..... but only some of those are common multiple values: 15, 30, 45, 60..... Clearly 15 is the smallest of the common multiples, thus the LCM. Thought, will the LCM of two prime numbers always be their product?

**Chapter 5 Discussion Notes**
 * [[file:MA224Ch5.pdf|My Notes]]**

**__Fractions__** Simply put, parts of a whole. The term **fraction** can be synonymous with the term **ratio**; both represent "undivided" quantities. The top number in a fraction is the **numerator** while the bottom is the **denominator**. If the value of the top number exceeds that of the bottom, the fraction is considered **improper**. Improper fractions can be written as **mixed numbers**, which consist of both whole and fractional parts. A fraction in **lowest terms or reduced form** means there are no common factors within the numerator and denominator.

To add and subtract fraction, there must be a **common denominator**. If one is not given, it can be created by selective multiplication. The process of finding a common denominator involves the number theory that multiplying by 1 has no effect on the value of a number [**Identity property of Multiplication**].
 * __Adding and Subtracting__**

__**Multiplying**__ The easiest operation to perform. Simply multiply numerators then multiply denominators. Reduce if possible.

__**Division**__ This operation requires some "blind faith" in the early years. To divide two fractions, the **divisor** (second term) must be **inverted** (flipped over) and the entire problem is then converted to multiplication.

Here is a great web site for elementary and middle school math teachers.[| IXL]

**Chapter 6 Discussion Notes** My Notes

**__Decimals__**


 * __Fraction to Decimal to Percent__**


 * __Scientific Notation__**

**Chapter 7 Discussion Notes**
 * My Notes**

**__Representing Data__** Graphing data is key to analyzing patterns in the data. The most common ways to represent data is with **Bar Graphs**, **Circle Graphs, and Line Graphs**. There are other forms that can be used such as the **Stem-and-Leaf Graph**, were each digit value of the data is represented as a "leaf" on the "stem" of base values.

__**Regression**__ is a measurement of how strong a graph imitates given patterns. Linear regression is measured from -1 to 1. A linear regression of 1 means that as each data value goes right, the corresponding value goes up the same from point to point. The graph of the data is a perfect line that is increasing to the right. Linear regression of -1 is similar, however, as data values go right, the pattern goes down in equal increments.

The measures of central tendency are mean, median, and mode. **Mean** is the average and is found by adding all data values together then dividing by number of data values. **Median** is the middle number in the set when the data is in order. **Mode** is the most commonly occurring data value. There is not always a mode, and there can be more than one.

__**Empirical Rule**__ 68% of all normal data falls within +/- 1 standard deviation of the mean 95% of all normal data falls within +/- 2 standard deviation of the mean 99.7% of all normal data falls within +/- 3 standard deviation of the mean

Any data value more than 2 standard deviations above or below the mean is considered unusual. Any data value more than 3.5 standard deviations above or below the mean is considered statistically impossible.

**Chapter 8 Discussion Notes**
 * My Notes**

**__Probability__** The probability of an event is the number of favorable outcomes divided by the number of possible outcomes. The probability of two events, A and B is the probability of A times the probability of B. [ P(A) x P(B)]. The probability of A or B is the probability of A plus the probability of B minus the probability of A and B. [ P(A) + P(B) - P(A) x P(B)]

**Chapter 9 Discussion Notes**
 * My Notes**